Research Laboratory
of the
Portland Cement Association
Bulletin 11
Shrinkage Stresses in Concrete
Part 1— Shrinkage (or Swelling), Its Effect upon Displacements
and Stresses in Slabs and Beams of Homogeneous, Iso
tropic, Elastic Material
Part 2 — Application of the Theory Presented in Part 1 to Experi
mental Results
By
GERALD PICKETT
March, 1946
( *m< AGO
Authorized Reprint from Copyrighted
Journal op the American' Concrete Institute
New Center Building, 7400 Second Boulevard
Detroit 2, Michigan
Jm. and Feb. 1946, Proceedings Vol. 42, pp. 165204 and 361400
Title 488 — o port of PROCEEDINGS, AMERICAN CONCRETE INSTITUTE Vol. 42
JOURNAL
AMERICAN CONCRETE INSTITUTE
(copyrighted)
Vol. 17 No. 3 7400 SECOND BOULEVARD, DETROIT 2, MICHIGAN January 1946
Shrinkage Stresses in Concrete*
By GERALD PICKETTt
Member American Concrete Institute
SYNOPSIS
Theoretical ex] :ttions of < and slabs
that occur during the course of drying and expressions for distribution
of the accompanying shrinkage stresses are derived in Part 1. These
expressions an on the assumption that the laws governing
the development of shrinkage stresses in concrete during drying are
analogous to those governing ..pment of thermal stresses in
an ideal body during cooling. Three cases are considered:
(a) slab or beam drying from one face <>!
(b) slab or beam drying from two oppo and
(c) prism drying from four fat
The applicability of the equations to concrete is considered in Part
2 (to appear ACI Journ It is shown that the
course of shortening of prisms is in very good agreement with the
theoretical equations and that from a test on one prism the shortening
versus period of diving of other prisms of the same mat. rial differing in
size and number of sides exposed to drying can be predicted with fair
accuracy if the differences in size are not too great. However, it is shown
thai the theory must be modified to take into account inelastic deforma
tion and to permit the supposed constants of the mat .rial to vary with
moisture content and size of sp if the theory is to be in agree
ment with all results on all types of specimen of a given concrete.
Various tests are described which, when used in conjunction with
the theory, provide a means foi g some of the more fundamental
properties of concrete and for predicting the performance of concrete
under some conditions in the field
INTRODUCTION
Concrete, like many other materials, gains or lost nth changes
in ambient conditions. With each change in water con 1 concrete
♦Received by the Institute. April 30, 1945.
1 Professor ot Applied Mechanic* Kansas State < oUege, Manhattan, kans.; formerly Portland Cement
Association Research Laboratory, Chicago.
(165)
i' 'Uiii i  i mi .,ii,., /.,,  , ,,„ m u , ,.,,,! ,,, ,,,„
I '" hrinl 01 •• ii I ■ re ul1 oi tin lum< In i
"• i I i ll " 1 maj hi, ii i the perforroai I tl m n U
• QUI , MM .1
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 ' '"' '"•• iblfl n ,,i, hum the typi oi ,,,i
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"" formula ...■ thai t] ,,,i , i ,, , ,,,,.,,„,. ,,,.,, ,,
eli 'I ,1 i obi ■ Hool • i i
I I bji Qfttu bete iven
Including "" ' ling i .ii, hardened oemenl pa h 
' ■■"■•<■•■ I "' h i "I mti i before hardening tend to .!«•
1 (hi have been n .1 1 obej Hook.
' Pi i" ''' 'i' und( i n i ,i train n  free i. , n
'■• '""i I in i. i.m be l i while the
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■■■'* drj i.ni tendini to brink wherea I ■ „
'■' "" ' h ""'" I tending I duel volume
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'"'•' h i to] te additional hydra
i°h Jiffe i i,,, ,, ln
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'•■' "■' I  lallj Indetermii
I I be! are off I „,,,.,. the
" B« proximal , .,,,,! with what
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'"' I *hl< i, ,,,i ,,,„
'"" '" ' I " 1 " 1 ll " 1 b 'rete would i bettei
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•" iblHtj toreheve tre b, , .,,,,.
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'"•" ■ ' .1 lanoj i.. .... ,.„„,„
' ''••' " '"•"•"■ i thedeflele ,,!.,,,,,
"' "hole pari i„ tended .
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„„';:,' ';; ,l "7 ■■ ^ i...
SHRINKAGE STRESSES IN CONCRETE 167
the distribution of shrinkage stresses in, concrete beams and slabs during
the course of drying.
Second, to show by means of data from specimens under controlled
conditions the manner and degree to which the equations apply to con
crete.
Third, to suggest methods for studying some of the more fundamental
properties of drying concrete.
No attempt will be made here to give a complete analysis of stresses in
concrete. In particular, the effect of aggregate particles on the stresses
within the hardened paste will not be considered.
Before expressions for shrinkage stresses in concrete can be derived,
assumptions must be made in regard to the relation between shrinkage
and moisture content and the laws controlling the flow of moisture in
concrete as well as the relation between stress and strain.
The actual relationships are not as simple as could be desired. If the
flow of water were entirely by vapor diffusion, if the vapor pressure of the
water in the concrete were proportional to the moisturecontent, and if
permeability were independent of the moisturecontent, then the differ
ential equation for the flow of water would be a partialdifferential
equation known in physics and mathematics either as the diffusion
equation or as the equation of heat conduction. Carlson, 1 * in a study of
distribution of moisture in concrete, assumed that this equation applies.
If the flow of water could be expressed by the diffusion equation and if the
shrinkage (or swelling) tendency! of each elemental volume were
linearly related to the moisturecontent, the unrestrained shrinkage (or
swelling) could also be expressed by the diffusion equation. This possi
bility was also considered by Carlson. 1 But the flow of water is different
from that indicated by the diffusion equation, and the relationship
between the change in moisturecontent and unrestrained shrinkage is not
linear as required by these equations. Moreover, satisfactory expressions
for either the flow of water or the moistureshrinkage relation have not
been found.
It is believed that moisture in concrete flows partly as liquid in capillar
ies, partly as vapor, and partly as adsorbed liquid on the surface of the
colloidal products of hydration. While drying progresses, the vapor
pressure of the water remaining in the region losing water decreases
progressively with the moisture content. This change in vapor pressure
with change in moisture content is not linear with respect to moisture
*See references at end of text of Part 1.
fBy shrinkage (or swelling) tendency is meant the unit linear deformation due to any cause other than
stress that would occur in an infinitesimal element if the element were unrestrained by neighboring elements.
It is not to be confused with the average unit deformation, commonly called shrinkage, of a socalled un
restrained specimen, nor with the resultant linear unit deformation which for the xdirection will be de
signated ?z. Hereinafter, the linear unit shrinkage tendency will be referred to either as unrestrained
shrinkage, for clarity, or merely as shrinkage, for brevity.
JOURNAL OF THE AMERICAN CONCRETE INSTITUTE January 1946
. ther i the rate <«f How proportional to the gradient id \
and relative proportions of the sp upied by
liquid : a drying proceeds. Thia fact, ae weX\ na the
nonuniformit a believed to be partly responsible for the
bich vapor pressure depends on moisture content and the way in
win ■■ gradient of vapor pressure.
ghtloea relation is different for different
ritioo of the concrete and the conditions
, Bferenl during first shrinkage from
subsequent volume changes, [fasatui
the ratio <>i change of length to lo
At first, comparatively Bmall cha
jiit . The higher the watercement
I be smaller the change during
' < ■ omes imi<li larger and
i i w bich ii may cither increase
j,i [t is believed that
■ bat bold 1 1n* \\ai. I e Bhape,
cles; and (c) th< 'It oi the
• red in any at udy
concrete !«■'
• «i.M'> i rom t In
hrough t he si •
^e> ImiI h
;.prc»\ima1i T\
l:i\\ tO In ;i — Mined, hilt
m, ill h<
and foi
1 \1<
.nd \\hil<
•
SHRINKAGE STRESSES IN CONCRETE 169
the diffusion equation applies to shrinkage even though the simple rela
tions that are implied by that assumption are contrary to fact. It is fur
ther assumed that concrete follows Hooke's law. The derivations given
in Part 1 are based upon these assumptions.
Since in Part 1 the derivations for deformations and stresses are based
on the assumptions that shrinkage follows the diffusion equation and the
material follows Hooke's law, the equations are even more applicable to
thermal stresses in metals than to shrinkage stresses in concrete. In fact,
much of the mathematical work given here was taken from the literature
on diffusion of heat and on thermal stresses, as the references will show.
However, certain corresponding coefficients in the two problems are of an
entirely different order of magnitude. For example, the numerical value
of the thermal diffusivity for steel expressed in square inches per second
is approximately the same as the numerical value of the shrinkage diffu
sivity of concrete expressed in square inches per day. Because of the
relatively slow diffusion of shrinkage the application of the hypothesis to
the shrinkage of concrete necessitates the study of early transient condi
tions (usually ignored in the treatment of heat).
PART 1— SHRINKING (OR SWELLING), ITS EFFECT UPON DISPLACEMENTS AND
STRESSES IN SLABS AND BEAMS OF HOMOGENEOUS, ISOTROPIC, ELASTIC
MATERIAL
Notation
S = free, unrestrained unit linear shrinkagestrain
~S = free, unrestrained unit linear swellingstrain
5 ro = final shrinkagestrain under fixed ambient conditions, value of S when t = a>
S av = average shrinkage over the volume of the specimen, the same as average shorten
ing per unit length if the material follows Hooke's law
t = time in days
k = diffusivity coefficient of shrinkage in sq. in. per day
/ = surface factor, characteristic of the material and the boundary conditions, in in. per
day
a, 6, c, d, I = distances related to the dimensions of the specimen in inches
B = fb/k, a nondimensional parameter
T = kt/b 2 , a nondimensional parameter
B c and T c , nondimensional parameters corresponding to B and T and used when a
second characteristic dimension of the specimen must be considered
x, y, z = rectangular coordinates
n = nth root of ptan = B
$ m = same as /3 n except used in connection with c, whereas n is used in connection
with b
A „ = Fourier coefficient
2B „ 2£,
F n = , F m =~
& + B + j3» Bl + B c + fil
b f„, a. A
$1 el
H n = — F m ff m = — F„
170 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE January 1946
V»A 2 J
<r t = normal components of stress parallel tox, y, and z axes— positive if
tensile, negative if compressive.
= elongations inx— , y— , and 2 directions
Txv, *». T r , = shearingstress components
y*,, 7*z, Ty» = shearingstrain components
E = Young's modulus in
u = Poisson's ratio
r = deflection in inches, displacement of the elastic line in the ydireetion
N = the normal to the surface directed outward
rX x=
*(*) =
2 rl
P(x) = — = j e fa t the probability integral
2 r 00 _ X 2
 m
fa = 1 — 
1
cos n 
cc 7^ COS 0* 
*< = 1   F m 
1 cos &»
H b
0= r«
1   #.
1
H. = 1
Equation for diffusion of unrestrained shrinkage
The diffusion equation is a mathematical statement of the fact that for
each infinitesimal volume  the excess of the substance in question
flowing in over that flowing out per unit of time is equal to the rate of
mcr ; ft <* in that volume. When similar assumptions are
k in regard to shrinkage, shrinkage thus being treated as if it were a
;ust as heat is so treated, the result is
'' La* 1 ty 1 az' J "
where fc is the diffusivity of shrinkage.
The equation becomes
..(1)
k 5 **—«
2
SHRINKAGE STRESSES IN CONCRETE 171
at exposed boundaries and
^= (3)
dX
at sealed boundaries
wherr
N is the normal to the surface, directed away from the body
/ is the surface factor
S oo is the value that S will eventually reach under fixed ambient
conditions.
Equations 2 and 3 correspond, respectively, to Newton's law of cooling
at exposed boundaries and to no flow of heal 3 erfectly insulated
boundaries in the analogous problem of flow of heat.
If the boundaries of the body are not parallel planes, a transformation
of Equation 1 from an expression in rectangular coordinates to some
other form is usually desirable. For i . if the bod si circular
cylinder, Equation 1 is best transformed to
Ldr* r dr r 2 d9 2 d z J dt
where r, 6, and z are cylindrical coordinates. Frequently, the condition
— = at some boundaries or some other conditions will make S inde
dX
pendent of certain coordinates and thereby simplify Equation 1.
Since the form of the solution for S depends upon the form of the
differential equation, the form of the solution is dependent upon the
boundary condition and the shape of the body under investigation. The
initial conditions (values of S at t = 0) and any variation in boundary
conditions with time will also affect the form of the solution.
Assumptions as to elastic properties
After a satisfactory solution for S has been obtained, then displace
ments and stresses will be found by the application of certain funda
mentals of the theory of elasticity. The solutions for stresses are here
restricted to homogeneous isotropic solids that follow Booke'a law.
Also, as will be brought out below, the effect of Poisson's ratio will be
neglected in some cas
Effect of shape of body on relative values of principal stresses
The state of stress at any point in a body is defined by the directions
and magnitudes of the three principal stresses. The three principal
stresses in wide labs and in narrow brain will be in the directions of
length, width, and depth, respectively, if the bodies are under uniform
exposure either from one or from two opposite face and are without
external restraint. The principal stress in the direction of depth (normal
•
1946
tresses will be
D  MBMfled t<» «i«*pt It the prinri
I
« VM.ltl
«  W ill Im
M ON
z
SHRINKAGE STRFSSES IN CONCRETE 173
Equation 2a is also satisfied if fi n is I he nth root of
(3 tan 13= f  (2b)
n
i.e., j8 B tan Pn = ~ (2c)
A
The above statements may be verified by substituting S from Equation
4 into Equations la, 2a and 3a.
For time t = » , Equation 4 reduces to S = S co , which is in accord with
the definition of 5 oo . An infinite series of terms such as the trigonometric
series in Equation 4 is necessary to give an arbitrary distribution of
shrinkage at time t = 0.
If the initial conditions are such that S = when t = 0, then the
Fourier coefficients A n are given by*
^n =
2 / " 6
O co tC
S n (fbV + Jg ,
U/ k
cos$ n (fb\
It therefore follows that
V
COS^
s sr t£ „ h
Si
where
= 1 Te'S./ (5)
£— cos p n
F = 2B
' n
5 2 + B + ft
Equation 5 (in slightly different form) and similar equations for other
shapes and other conditions, applied to analogous phenomena, may be
found in the literature of mathematical physics such as the textbooks
of Byerly, Carslaw, and Ingersoll and Zobel. Various tables and diagrams
have been prepared from which the numerical relationship of the four
nondimensional quantities S/S «, y/b, B, and T may be found, such as
Fig. 4, page 841 of Perry's Chemical Engineer's Handbook (1934),
* T he general procedure of obtaining Fourier coefficients to satisfy initial conditions semewhat analogous
tO tin present problem ifl given in Articles 66 to 68 of Byerly'a Fourier Series and Spherical Harmonics
on: Ginn A Co., 1893).
174
JOURNAL OF THE AMERICAN CONCRETE INSTITUTE January 1946
To use more than a few of the terms in Equation 5 for the evaluation of
S/S » is very laborious because of the difficulty in evaluating ft, and F n .
The number of terms required for a given degree of precision will depend
somewhat upon the parameters B and y/b but is chiefly controlled by the
parameter T. Computations show that very little error is introduced by
neglecting all terms in the series except the first if T is more than about
0.2; but many terms are needed for the usually desired precision if T is less
than 0.02, — the smaller the value of T the greater the number of terms
needed. Very precise values of S/S oo for small values of T may be found
without the tedious computation indicated in Equation 5 by using
another expression which will now be derived.
Solution in terms of the probability integral As long as the shrinkage at
the sealed surface remains negligible, the distribution of shrinkage from
the exposed surface inward will be nearly independent of the distance
between the two surfaces. Suppose that instead of considering the surface
at y = to be sealed, the body is considered to be extended to infinity in a
negative wdirection. Then instead of the boundarv condition — =0 at y
dy
= 0, the requirement will be
S = (6)
at y = — oo .
The solution* satisfying Equations la, 2a and 6 and giving S = when
t = Ois
S_
Sc
=
[1Jj
[>?]
_2Vf .

_ 2 V T _
B(l
) + B*~T
■ {')
where <j> (x) is — = / e dx and T is again used in place of — .
The quantity 1 — o ■ ■■
4 f
dx, is known as the proba
bility integral. Values of <t> [z) may be readily found by using a table of
the probability integral.
Numerical calculations show that Equation 7 gives values that differ
from those given by Equation 5 by an amount less than the value of
S/*Sa: at y =0; therefore, Equation 7 may be used in place of Equation 5
whenever T i> so small that S Sm at y = is less than the permissible
error.
*Thi*> solution it very simiLar to that given for an analogous problem bv Carslaw in Article 25 of The
Maciiiillan <k Co.. Ltd., 2d e<l., 1921).
SHRINKAGE STRESSES IN CONCRETE 175
Table 2 and Fig, 7, showing S/S « in terms of y/b, kt/b 2 , smdfb/k, were
prepared from Equations 5 and 7.
Stresses and strains
Continuity, Hooke's law and equilibrium. As stated previously, the
solutions for stresses are here restricted to homogeneous, isotropic solids
that follow Hooke's law. Equations for the stresses that would be pro
duced in such a body by the shrinkages given by Equations 5 or 7 will
now be derived.
The shrinkage S has been denned as the linear unit deformation that
would occur if each infinitesimal element were unrestrained. However,
the properties of a continuous solid will not permit an arbitrary distribu
tion of deformations ; therefore, unless the distribution of shrinkage given
by Equation 5 happens to be compatible with the conditions of continuity,
stresses will be produced that will modify the deformations so as to make
them compatible. Although in general six partial differential equations
are required for a complete mathematical statement of the conditions of
compatibility, 3 these are reduced to
** (8)
dy 2
for either long narrow beams (plane stress) or slabs (plane strain) if the
stresses are considered to be independent of the longitudinal coordinate x.
The term e x is defined as the resultant unit deformation in the zdiree
tion (the direction of length). It is therefore the algebraic sum of shrink
age, S, and the strain produced by stresses. a v is obviously zero; and if
Poisson's ratio is zero or if the discussion is confined to narrow beams, a t
is negligible. Therefore,
e x = &S (9)
E
or, solving for stress,
<r,  E(e t + S) (10)
where E is Young's modulus,
The restriction imposed by Equation 8 requiring that the expression
for longitudinal deformation contain no terms in y other than the first
power (second derivative equal to zero) is equivalent to the assumption
usually made in the elementary theory of beams that "plane crosssections
remain plane." If longitudinal restraint is complete, then e x is zero and it
follows from Equation 10 that <r x = ES. If, however, longitudinal short
ening is permitted but complete restraint against bending is provided,
then e x is not zero but is still independent of y. If the beam has no external
restraint, the nonsymmetrical distribution of shrinkage causes it to warp,
making e x a linear function of //. For no external restraint the equations of
176 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE January 1946
equilibrium (summation of forces in the jdirection equal to zero and
summation of moments about the zaxis equal to zero) become
/
<j z dy = 0.
(11)
and
/
<*xV dy =
(12)
It may be shown by substituting Equation 10 into both Equations 11 and
12 that if shrinkage (S) is either independent of or a linear function of y
an unrestrained beam will be free of stress (e I =  S and c x = 0). For
any other variation of shrinkage a stressed condition must result because
the restriction on e x (Equation 8) will not permit it to be equal and oppo
site to S if shrinkage is a nonlinear function of y.
The only solution for e x that satisfies Equations 8, 10, 1 1 and 12 is
b b
= ( 6 v 4 )i/ 5 ^ + ( 6  12 i)i /■**■<*>
When this value of e x is substituted into Equation 10, the result is
= E
o o
(14)
Finally, 5 from Equation 5 may be substituted into Equation 14 thus
g Tf, J 1 " 6 ! 8 /, 11 * n u *"" ;'»""> »>eam as a function of the parameters
V, o, kt/b,fb/k, 5 c and of Young's modulus. This substitution will not
be made until later, because it seems advisable at this time to consider
another approach.
Solution by sup, rposition. Although the above derivation is short and
is in the simplest form for checking the mathematical correctness of the
equation, a derivation in which elementary solutions are superposed is also
desirable because it will be easier in general to understand and because the
hnal _ expresMon, are in more usable forms. In this second derivation the
resultant stress ... U cowddered as consisting of three parts. The first part
s that stress which would be produced by complete restraint against
longrtudmal deformation; the second part is a uniform stress equal to and
opposite m s,gn to the average of the first part; the third part is a stress
SHRINKAGE STRESSES IN CONCRETE
177
resulting from a simple moment that is equal to and opposite in sign to
the moment produced b\ r the sum of the first two parts. That is, the first
part alone <j x f would result from complete restraint, the sum of the first
and second parts <r/ would result from restraint against warping only;
the sum of all three parts, i.e., a XJ would result if no external restraint were
applied during shrinkage.
Although in this derivation an expression for <r x appears to be the
ultimate goal, expressions for a J and for a/ are also desirable. The stress
trj may be representative of the stress in pavement slabs or building
walls that are restrained from shortening and the stress <r/ is representa
tive of an unrestrained wall drying equally from two opposite sides (Case
II discussed later).
Since for complete longitudinal restraint e x = 0, it follows from Equa
tion 10 that the first part of the stress is
aj = ES (15)
i r
Since the average value of trj is — I aj dy, the second part of
b
e r
a x is — r I S dy; therefore, the sum of the first and second parts
(<r/) is given by
a x " = E
\f
Sdy
• (16)
The moment produced by <r/ is the moment necessary to prevent warp
ing. This moment per unit width of beam is found by multiplying Equa
tion 16 by y dy and integrating. This gives
i b
M
j *; ydy = E J Sy dy  J I
Sdy
L
.(17)
For no external restraint this moment must be removed by superposing
an equal and opposite moment. The stress resulting from a moment — M
is given by the elementary theory of beams as
M {y  6/2)
1/12 ¥
JOL. CONCRETE INSTITUTE January 1946
T 1
I / /
L I
.•s N t!nT
i
■ , ., I V
/ Ir
Ofllhff'l

■ teaM
SHRINKAGE STRESSES IN CONCRETE
179
in terms of the deflection iw it produces, the expressions for the stresses
are put into more usable forms. When this is done, the following equations
for stresses are obtained.
For complete longitudinal restraint (first part of <r x ),
a x > = ES (15)
For restraint against warping only (sum of first and second parts of
Oi
<r x " = E (S  S av ) (20)
For no external restraint (sum of all three parts of <r»),
/ y\ 26^
• (21)
Evaluation of the parameters ~ and — ^
When Equation 5 for
S/Sc is substituted in Equation 18 for shortening and in Equation 19
for warping and the indicated integrations are performed, the result is
b °°
1
.(22)
2b Vrri
3l*S<
i r s s av >y
= bj sZ ydy " 25. = 2
rft
e G.
(23)
where
and
H n =
2B 2
G,
01 (B* + B + fil)
\cosB n 2 / 81
If T t the nondimensional timefactor, is small, the series in Equations
22 and 23 converge rather slowly, and in that case it is convenient to use
the following equations obtained by substituting Equation 7 into Equa
tions 18 and 19, respectively.*
Sa
e i <t>(B V T)  1 + =
.(24)
The lower integration limit for each integral is decreased from to — »;
180
JOURNAL OF THE AMERICAN CONCRETE INSTITUTE January 1946
2bv„
WS<
JL I_
2B + B*
v*
r>...(25)
Furthermore, if the parameter B V T is very small, it is still better to
use the following equations obtained by expanding the expressions in the
brackets of Equations 24 and 25.
^~ = BT
4 _ 1 _
■= B V T +~(B V TY
2bv n
3Vt
V BT
15 V
8 ,—
(BVr) 3 +
3Z 2 S,
(24a)
flr / b\/ 4 _
4 ^^i^v *■>•)
. (25a)
In general the following rules will be found applicable for rapid evalua
tion of the parameters = and ^t to a fair degree of accuracy.
0<x> Ot^O m
If T is more than about 0.05, use Equations 22 and 23.
If T is less than about 0.05 and B is more than about 5, use Equations
24 and 25.
If T is less than about 0.05 and B is less than about 5, use Equations
24a and 25a.
Forces and moments necessary for complete restraint. The force neces
sary for longitudinal restraint is JV dA. Therefore, the average force
b
per unit area is jJ,,' dy. From Equations 15, 5, and 22 this becomes
force per unit area = ES t
2
e II.
■ (26)
From Equations 17, 5, and 23, the moment per unit width necessary
tor restraint against warping is found to be
°° Tfi
U = £Sc & ^ e G n
1
(27)
SHRINKAGE STRESSES IN CONCRETE
181
Simplification by taking B as equal to infinity. The principal equa
tions derived above reduce to simpler forms and the computation of
numerical values is less tedious if the assumption is made that B, i.e.,
fb/k } equals infinity. If B is large, say 100 or more, the error introduced
by assuming it to be infinity is negligible. However, if B is less than
about 5, the error introduced by considering it to be infinity may be
appreciable as is shown, for example, by Fig. 8, 9, and 14. Whether
justifiable or not, the assumption that B = oo is frequently made in anal
ogous problems to which the diffusion equation applies. This assump
tion was made by Terzaghi and Frohlich 4 in developing the theory of
settlement of foundations due to consolidation of underlying material, by
Glover 5 in a study of distribution of temperature in concrete dams, and
by Carlson 1 in a study of distribution of moisture and shrinkage in con
crete. The more important of the above equations for the special case
of B = cd are given below :
Equation 5 becomes
S ^4(l) 1 (2nl)*jT
cos (2n  1) — .
s
(2fl  l)ir
2b'
Equation 7 becomes
Equation 22 becomes
Sa
Sc
__8 s 1_
" 7T 2 Z. (2n 
 (2h  I) 2 (7r 2 /4) T
I) 2
Equation 24 becomes
v J T
O oo \ 7T
Equation 23 becomes
2bv m
WSc
= z
7T 2 (2n  \y
i
Equation 25 becomes
2bv ma:
3l 2 Sc
yr .
w (2n  1)
 1
_ (2n  I) 2 — T
4
T .
182 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE January 1946
Tables, curves, and computations.* Tables and diagrams such as those
by Newman 1 are available from which values of N S« and S m /Sd may
be determined. However, such published tables arc in general not ade
quate for the present problem. The smallest value of the parameter T
used by Newman in his computations was 0.1, whereas the stresses in
cone sired for a much earlier period. The tables given
here were prepared for T as low as 0.001. Moreover, so far as is known,
er — ; not previously been evaluated fortius or any
analogous problem and its bion is necessary for the problem here
■ oi Equations 15, 20, and 21 for the theoreti
one side under the different modes of
bh i luce quantities ■ , ' ; and 
• ion ol ih« three parameters y b, B and
I ig 7 Tin' second quan
Ild V i given in Table 'A and shown
ni:i!)til \  '" B£ a function of B and
• >/ ■ . s
M\ in Fig 9. Tables . r > and 6
20 and 21 as functions of i be
prepared after the three
1 abl<  2, •'». and 1 1 Results foi
0, li. L2, and 13 Fig. l i shows
rsufi the parameb
on <»i the tables and diagrams
iaf ions based upon the
i c used instead ol 1 lie
mtered. I mple,
q 7 then
l ) ( u)i
1.1/ 9.1 /
• • »'•' ' " ■■•'■<■> ' .. ..:... ...:.'..  ,„••.! c i' » *r :>!,!.► *.»♦ :,, J COliM ;.U:\
SHRINKAGE STRESSES IN CONCRETE 183
From mathematical tables
i op;
4> (1) = 0.15730; 4> (1.5) = 0.03389; e = 3.4903
Therefore S/Sa> = 0.1573  0.03389 X 3.4903 = 0.0390.
Note that this is the value given in Table 2 for the above values of
B } T, and y/b. Also note that for the same B and T the table gives zero
for y/b = 0, showing that it was permissible to use Equation 7 instead
of Equation 5.
When the theoretically correct equations are used, the computations
are more involved. For instance, let T = 0.1 instead of 0.01 in the
above example. T will then be so large that S/Sa> will have an appreci
able value at y/b = 0. Therefore, Equation 7 will not be applicable and
Equation 5, the exact equation, must be used. A substitution of values
for T and y/b into Equation 5 gives
00 —QlQl
A = i _ V F cos ° 8 ^
So* 2i ( " co$p n
1
The first step in evaluating the above expression is to determine n
which Equation 2c shows to be a function of fb/k and n, i.e., B } and the
integer n. The determination of 3 n by interpolation is simplified by the
introduction of a„ where a n depends on B and n. The equation for 0„ is
then written
n = (n  1 +a n )7r (28)
Curves of a n versus B for the first six values of n and for n = 21 are
shown in Fig. 2. By means of Fig. 2 and Equation 28 any desired j3 n
may be found with reasonable accuracy for any value of B, The first six
values of n for several different values of B are given in Table 1.
After finding n for the given values of B and n, the factors F nt
cosp n , cos (fin) and e ~ T &* are determined. F n and cos /3 n as func
tions of B are shown in Fig. 3 and 4, respectively, for the first six values
of n. The functions cos i & n \ and e ~ l $ n are readily obtained from
mathematical tables after the products 0« ~ and T &l have been deter
b
mined. When the proper values of the four factors listed above are sub
S
stituted, the above equation for^— becomes
O as
184 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE January 1946
0.84147 X 0.3152 X 0.4966 _ 0.1965 X 0.2161 X 0.9963
O oo
0.2541
0.00844 X 0.1286 X 0.7280
0.8101
0.6277
This reduces to — = 1  0.5183  0.0674  0.0009 = 0.4134
The values for S/S » given in Table 2 were computed by one or the
other of the methods illustrated above
In preparing Table 3 from_which Fig. 8 was constructed (Fig. 8 shows
shortening as a function of V T for various values of B), values of H n in
Equation 22 were needed. Values of H n as a function of B for the first
six values of n are shown in Fig. 5. In like manner, Fig. 6 showing G n
as a function of B, served in the preparation of Table 4 from which Fig
9 was constructed. Of course, for small values of T, Fig. 5 and 6 are not
necessary since either Equation 24 or Equation 24a is used instead of
Equation 22 and either Equation 25 or Equation 25a is used instead of
Equation 23, depending on the value of B.
Application to beams or slabs of any widthtodepth ratio when Poisson's
ratio ts not zero. The effect of Poisson's ratio was neglected in the pre
ceding derivations. Its effect stated in general terms in the introductory
remarks in regard to Case I will now be analvzed in more detail. If
Poisson s ratio is not zero, Equation 9 for e, and Equation 10 for c x will
be modified to include the effect of c t . That is,
'*w. = E(e. + S). (10a)
However, if the ratio of width to depth is small, „ z will be negligible and
■ tion 10a reduces to Equation 10. On the other hand, if the ratio of
width to depth is very ] ar ge (a slab), the width being comparable with
tlKJe^th, then a. will be equal to „,. If „, = ,„ then Equation 10a
°' = T~ (e * + V (10b)
t J'Tn? 1 ' difference u between E^on 10b for a wide slab and Equa
.on 10 or a narrow beam is the factor * which occurs in Equation 10b
bul ,,„ . Equation 10. Therefore, for stresses in a slab, E k ! Equations
15, 20, and 21 is replaced by £.. The stresses in beams „ dth
tod..p,h nmo i, in„.r„K,lia, ( . will have stresses intermediate between
of narrow beams and of slabs. Since ,o, appear in Equa
SHRINKAGE STRESSES IN CONCRETE 185
tion 22 for average shrinkage nor in Equation 23 for warping, these
quantities are the same for narrow beams and wide slabs.
CASE II— SLAB OR BEAM DRYING FROM TWO OPPOSITE SURFACES
Equations taken from those derived for Case I. Since the flow of mois
ture in a slab drying from only one surface is believed to be the same
as that in either half of a slab of twice the depth drying from two op
posite surfaces, it will be assumed that the theoretical equations de
rived for shrinkage of a beam or slab drying from only one surface will
apply equally well for either half of a beam or slab drying from two
opposite surfaces. The plane midway between the drying surfaces will
be taken as the plane y = as shown in Fig. 1 for Case II. Since the
two halves of the beam will mutually restrain each other from warping,
the equations for stresses, strains, and shortening in each half will be the
same as those given previously in Case I for a beam restrained against
warping and drying from one surface.
CASE HI— RECTANGULAR PRISM DRYING FROM FOUR FACES
Shrinkage
The differential equation and boundary conditions. For a prism drying
from four faces but not from the ends the diffusion equation reduces to
\dy 2 dz 2 / dt
The exposed faces of the prism will be taken as the planes y = =*= b,
z = ± c, as shown in Fig. 1 for Case III. The boundary conditions then
become
^ = ±L(Sa>  S) (2a)
dy k
at the boundaries y = =*= b and
3? = *£(fl.  S) (3b)
dz k
at the boundaries z = =*= c.
The solution satisfying Equations lb, 2a, 3b, and giving S = at
t = and S = S*> at t  °o is the following:
m cos A^ir"  r «* cosfij
O oo
I
COS f} n
1
Ye F m c
cos p„,
. (5a)
1
where p m F m} and T c correspond to p n , F n , and T, respectively, the differ
ence being that the dimension b has been replaced by c.
186 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE January 1946
Shrinkage expressed in terms of the solutions given for a prism drying
from one face or two opposite faces. Since the infinite series in the first
bracket in Equation 5a is identical with the one given in Equation 5 and
the infinite series in the second bracket is like the first except that y is
replaced by z, b by c, etc., and since Equation 5 applies to either half of
a slab exposed on two opposite surfaces (Case II), it follows that the
brackets have the following values:
'.   = 1  & ,
2e
■T e &
cos f} n
cos m 
c
cos fi m
(29)
= 1  *
.(30)
where </> b is the value S/Sa> would have if only the surfaces y = ± b
were exposed and <f> c is the value S/Sa> would have if only the surfaces
z = =±= c were exposed.
A substitution of Equations 29 and 30 into Equation 5a gives
5— — 4>b + 4>c
O 00
<t>t>4>c
.(5b)
Equation 5b shows that the evaluation of shrinkage for a prism dry
ing from four surfaces becomes a problem of adding the independent
effects of drying from surfaces that are perpendicular to each other and
then subtracting a term proportional to the product of the separate
effects.
For example, consider the shrinkage tendency at the point y = 0.46,
2 = 0.8c in a prism for which c = 26 (width equal to twice the thickness)'
Let/, k, and t be such that/6/fc = 5.0 and kt/b* = 0.20; then fc/k =
10.0 and kt/c* = 0.05. </> 6 is found in Table 2 or from Fig. 7 to be 2398
Since Table 2 was prepared for B equal to 0.1, 1.0, 5.0, and » only, and
since fc/k = 10, 0, can be found from Table 2 only by interpolation
However, examination of Table 2 indicates that for kl/c* equal to 05
Equation 7 can be used instead of Equation 5 without appreciable error
and therefore the equation rather than the table will be used to obtain
<i> c  From Equation 7
=  ' r  ~ * (■ °^— + 10 V0.05^
V2V0.05/ \2V0.05 /
2+5
.2V0.05/ V2V0.05 /
From tables giving probability integrals and the exponential function
4>c = 0.5273  0.000156 X 1097 = 0.355
Therefore at y = 0.46, z = 0.8c, and t = 0.20 b,'k
s
— = 0.2398 + 0.355  0.2398 X 0.355 = 0.510.
SHRINKAGE STRESSES IN CONCRETE 187
Shortening expressed in terms of the shortening of a prism drying from
one face or from two opposite faces. The average shrinkage S at is given by
b c
s av = ~tff Sdy " : (18a)
From Equations 5a and 18a
r# ro T e 0L
5fL_ = i  5" e H n V e H m (22a)
1 1
or
^ « H h + H c  HJI* (22b)
O CO
where H b is the value S S would have if only the surface y = ± b were
exposed and // ifi the value S ar /'Sao would have if only the surfaces
z = ± c were exposed*. Therefore, the average shrinkage, and conse
quently the shortening, if the bod ic, of a prism drying from four
sides may be found by considering the separate effects of drying from
opposite ides in pa;
For example, consider the shortening of the prism discussed above.
H h is found in Table 3 to be 0.3510 and H c is found to be 0.1753. Therefore,
— = 0.3510 + 0.1753  0.3510 X 0.1753 = 0.4 7
O QD
Stresses and strains
Nature of the problem and the method to be used to obtain a solution. In
■■s I and II previously discussed, where shrinkage was a function of
time t and only one space coordinate y and where the problem was further
simplified by neglecting the effects of the length and the width of the
imen on the distribution of stresses, a solution for the one stress
involved was readily obtained. However, in the problem now under con
sideration, a prism doing from four surfaces, shrinkage varies with an
additional coordinate z. As a result stresses vary with this additional
coordinate also, and more than one stress will be involved. The problem
will be somewhat simplified by neglecting the effect of the length on the
distribution of st e., the assumption will be made that stresses do
not var> along the length. The distribution of given by the
solution based on this assumption will deviate a negligible amount from
•If a prism were drying from all six surfaces, the corresponding equation would be SWS. = H. + Hh
— He  IUH  //// f  HH* — HaHiH,. Another way of expressing these relations is given by Glover
VRdLI
188 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE January 1946
the theoretically correct distribution when stresses in the central portion
of a long prism are under consideration (principle of Saint Venant 8 ).
The solution for shrinkage in terms of two space coordinates was air.
when only one coordinate was involved (Equation 5a for
I II compared with Equation 5 for Cases I and II j, because tendency
to shrink is considered to be a scalar quantity. On the other hand, since
stresses are tensor quantities, the solution for stresses usually becomes
much more complicated whenever more than one coordinate is involved.
In fact, elastieians have obtained exact solutions meeting all boundary
conditions for only a relatively few problems in which stresses were func
tions of at least two coordinates and then only by considering the body
to be infinite in the direction of one of these coordinates. The difficulty
is that since stresses are tensors, boundary forces are vectors, and in
twodimensional problems two components of force must be satisfied at
each boundary. The specified conditions of stress at anv two opposite
boundaries can be satisfied by superposing particular solutions of the
differential equations in accordance with the usual methods of Fourier
analysis. But, in general, solution* ing rigorously the boundary
requirements at two pairs of opposite boundaries simultaneously cannot
be found by the usual metho
A method of solving such problems after the appropriate differential
equations have been derived was explained by the author in a recent
paper*. That method will be used here. It is about the same as that u*ed
previously by Taylor" and by Timoshenko" in analogous problems.
Der ' the differential equations rehr , shrinkage Bv
neglecting the variation oi and strains along the length of a body
the problem becomes a twodimensional problem in plane strain The
flowing equations taken from the theory of elasticity are then applica
Equations of Equilibrium:
<>y dz
+ a^ =0
Condition of Compatibility:
Modified to Include Isotropic Shrinkage:
dz 2
' ~ E k ~ ^ 9 " ^  S "  s  in ;hi ^ problem
_
SHRINKAGE STRESSES IN CONCRETE 189
€ v = — [— v.<*x + <r v  M°"z] "
 5
e g = — [ — /x^r — mc v + ^] 
E
 s
2 (1 + M )
7 U * = T M
E
The above seven equations giving relations between the eight unknown
stresses and strains can be reduced to the following two equations by
eliminating the four strains and the shear stress:
<r M = M (<r ¥ + O +E(S S av ) ■ (31)
y »(,, + a: ) = JL_ V *S (32)
1  n
where V 2 is written for — H .
By* dy 2
These two equations together with the two equations of equilibrium
and the boundary conditions that
fri
and r yz = at y = ±6
V «" 1,J  '&z
cr z and T yz = at z = =*= c
and Equation 5a for S constitute the mathematical statement of the
problem.
Solution for stresses. In general the stress a x will be larger than either
ff v or ov The stress r vz will be relatively small in all cases. If only the
value of the theoretical maximum stress is desired, a fairly good approxi
mation can be obtained by the following simplified formula:
<r x = E (S  S av ) approx (31a)
where S is given by Equation 5a and S av is given by Equation 22a. If,
however, an accurate theoretical value of all stresses is desired, a complete
solution must be obtained and this is given below:
The solution for <r yt <r zy and r yz meeting all of the above requirements is
as follows:*
r 2 _ q 2 2 b ^ *r~ An
*»= A °lzt+~c Z Z f^sa iy cosy ]Z
t=i i=i
♦Equations 33. 34. and 35 for stresses and Equations 36 and 37 for the coefficients B, and Ci given here are
alnWt the same as Equations i, 2, and 3 for stresses and Equations 6 and 7 for the coefficients B n and A n
MW*y4?&ento&L%™ The Aperies and the Antenna enter into the equations, given here in place
of the S™?n the stress S given there; otherwise, except for slight differences in notation, the correspond,
in^ Pauftfon^ are identical and the equations given here may be derived by the procedures given there.
Equation 38 for X? and Equation 39 for A may be verified by substituting Equations 33 34, and 5a ».
t Equa^on32andth e np^^ resulting equation within the domain
under consideration by the usual Fourier analysis.
190 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE January 1946
, \ r, cos yjZ r • . /
2 B >' T^Th ™ y SUlh ™ ~ ( 1+ y* b coth y& cosh y 3 y]
i=i
cosh y } b
Z r cos dig r
cosh ^ a%Z + ^~ aiC €oth a < c ) cosh Oiz] . . (33)
, & 2 3^
00 oo
,_i, l^ + i Z Z 7T C0S ^
cos 7 ; z
»*=1 3 = 1
Z„ COS 7,2 r . , .
'' c^sT^i biy smh ™ + (1  T ' 6 co ^ ^ 6 ) cos * r,y]
i=i
\ r cos a f y r . ,
Z 'cosh af [aiZ Slnh aiZ ~ C 1 + a ' c co ^ «.<) «wA a, 2 ]
(34)
t'=l
oo oo
> — r sin any sin y,z
00
4 'S" R Sin y > 2 I l ,
Z *' cos* 7i 6 ™ 6 c<rfA ^' 6 s '^ *»  7,2/ cosA 7/ y]
i=i
00
Z Ci cosA a,c LttiC C0 ^ a ' f s ''' i/! a ' z ~ «* cosh afi] .
1=1
■ (35)
where
a, =
iw
b '
T/2
_ _c_ AoJiy , 6
6 7T
F+}I«tz<>^
te«A l =
B,=
i'=l
i= 1
i^io
i +
*(<
m
c,
c
)
(36)
SHRINKAGE STRESSES IN CONCRETE
191
c A.(iy c
(iy ~  y (_i)
1" £■ —
c, =
i=i
tan/i —
i=i
i +
(S)'
B<
6 V
h^fcott™
to llll
(37)
.1., =
" = 1 '" = 1 v"*«") v "ft"/
n = 1 m = 1
The above equations, together with Equation 31 for a SJ constitute a
complete solution on the basis of the given assumptions.
In general, if both the parameters T and T c are equal to or greater than
0.1, the series given above converge very rapidly so that only a few terms
need be taken for a good approximation. The example given below will
demonstrate the use of the above equations.
Example: Stress at the middle of one side of a square prism for which B
equals 5 and at a time for which T equals 0.1. If only one term in each series
is used, the following values are obtained: From Equations 38 and 39
and Table 1.
ES " (0t + 0i)e Olfl.O 1 * //
A =
A.
1M
ES c
(1.3138* + 1.3138*) 0.8415* X 0.9130*
.4. =  2.0377
i<:s
1
ES<» (1.3138* + 1.3138*) 0.8415 2 X 0.9130 2
An — ~ — — " — : — / _? \2
(1  *0 **
(l  _£)'
V 1.3138V
192
A n =  0.0093
JOURNAL OF THE AMERICAN CONCRETE INSTITUTE January 1946
E a oo
lM
When these values are substituted into Equation 36, the result is
(_ 2 _^ +0.0093) ^'«5*£ Cl
B l =
1 + 7T (cotf* 7r — /an/i 7r)
or since Ci = B Y (square prism),
C, = B,= ~ 0.1472 —
When the above values and z = c, y = Q are substituted into Equation
33, the result is
+ 0.0093  0.1472 X 0.3583 + 0.1472 X 0.97731 ^*
Jl M
= p.0377
L 6
2 =C
y =
■[
+ 0.3396 + 0.0093  0.0527
r 0.1439 
■ = 0.4401 «
z = c
2/ =
1M
When the summation of each series is carried to two terms, the result is
ES
0,4214  — — and when the summation is carried to four terms in each
1
ES :
series the result is 0.4221 t^Jfl f or this stress.
1 M
The above shows that the series converge very rapidly for this example.
The stresses a t and r vz are obviously zero at the point under considera
tion.
From Equations 31, 5b, and 22b, Tables 2 and 3, and the above result
for c y ,
= [jj 0.4221 + (0.0221 + 0.6913  0.0221 X 0.6913)
 (0.2186 + 0.2186  0.2186 X 0.2186)1 £S«>
z = c
y =
Li  M
0.4221 + 0.3087 ES
]
2 = C
y = o
For all values of Poisson's ratio M less than 0.212, the above expressions
will give larger values for <r y than for *„ This fact is of interest because,
5E STRESSES IN CONCRETE 193
in bi 

OTHER METHOOS
D tO

■
I
M
SUMMA
I
ng, warp
I
r beam & na one ^ ntaina an
 n the correspond!!*
194 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE January 1946
equation for a slab or beam drying from two opposite surfaces. Com
putations show that the equation with the added term gives much less
stress tor the same size of body and the same period of drying. Compare /
Tables 6 and 5. " '
Equations are given for all the stresses in a prism drying from all four
surfaces, the third condition treated. These equations, though rather
complicated in appearance, can be readily evaluated if the desired accur
acy is such that only a few terms in each series need be considered. For a
rather rough approximation of the stress that is usuallv the most impor
tant in this third condition, the comparatively simple equation <r, =
E(S ~ Scv) is recommended.
Tables and curves are given from which the theoretical shrinkages,
stresses, etc, may be obtained, at any point in the specimen after any
period of drying, for various values of the physical properties, diffusivity
surface factor, ultimate shrinkage, and dimensions of the specimen.
Examples are given showing how numerical values mav be computed
from the equations and how the tables and curves may be used.
REFERENCES
JaLFe?. m g 7 S &\ ?e ^,^T27 COI1Crete MeB ^ n '" »* * "■ Carlson, ACI Jolr.val,
^^^'^^^^^^trst^hk and E  s  Sokolnikoff '
3. Theory of Elasticity, by S. Timoshenko, McGrawHill. New York, 1934, p. 196
Fran, S^^S^^St^" * * * *«*» "* °. K ' ™**
<rfCtenri^a?"'^ T ^ e GIo^ D ArT i T U,i0n in a v SucC ^ ion 0{Ulis Due t0 ReI ^
« "TkrT * „ OIo%er ' ACIJovR * A L. ^v.Dec. 1937,Proc.. V.34, p 105
In£. 0f T C h heS O Enl., i^ P g Tl9 S 31 lidS ' , ' b> ' * * Xwman ' Interim Publication, Am!
"Flaw of Heat in DanV' by R. E. Glover, ACI Journal. Xov,Dee. 1934, Proc.
8. Reference (3), p. 31.
pp 222^" ° fPtaUS ^ SMh  ** S  Timoshenko, McGrawHill, Xew York, 19M,'
12. Ref. (3), pp. 2123.
C^^bfati^p t JX%^^S^S!' T^T? 1 fi*— » Concrete
14 "A «JLi ArTi j , 1 ^ r ° c ^ m  • ,Jf Testing Matenals, V. 39, 1939, p. 913.
ss?" "*£& ass ra/raaarafstat
SHRINKAGE STRESSES IN CONCRETE 195
17. Relaxation Methods in Engineering Science, by R. V. Southwell, Oxford Univ.
Press, 1940.
18. "A Lattice Analogy for the Solution of Stress Problems," by Douglas McHenry,
J. Inst, of Civil Engrs., Dec. 1943, pp. 5982.
19. "Numerical and Graphical Method of Solving Two Dimensional Stress Prob
lems," by H. Poritsky, H. D. Snively, and C. R. Wylie, Jr., J. Applied Mechanics,
June 1939, V. 6, No. 2, p. A63.
20 "The Effect of Change in MoistureContent on the Creep of Concrete under a
Sustained Load," by G. Pickett, ACI Journal, Feb. 1942, Proc. V. 38, p. 333.
21. "Die Beeinflussung des Schwindens von Portlandzement durch Sulfate," by
G. Haegermann, Zement 28 (40) 599 (1939).
22. "A New Aspect of Creep in Concrete and Its Application to Design," by Douglas
McHenry, Proc. Am. Soc. Testing Materials, V. 43, 1943, p. 1069.
23. "Attempts to Measure the Cracking Tendency of Concrete," by R. W. Carlson,
ACI Journal, June, 1940, Proc. V. 36, p. 533.
24. "The Influence of Gypsum on the Hydration and Properties of Portland Cement
Pastes," by W. Lerch, unpublished.
25 "The Dependence of the Shrinkage of Portland Cement on Physical and Chem
ical Influences," by H. Kuhl and D. H. Lu, Tonind. Z. 59 (70) 843 (1935).
196 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE January 1946
m
• 7T~"
3_
Fig, 1 — Illustrations of the con
ditions treated. Shading indi
cates sealed surfaces. Case I
— Beam (or slab) drying from
one face only. Case II — Beam
(or slab) drying from two op
posite faces. Case III — Prism
drying from four faces. The
ends of the prism at x = * a
are sealed.
OSOp
0.45 
040 
OiS 
050 
n 0« 
020 
o ig 
010 
005  /
00
»here (J n is the n?> root ^^
* P n ton(i n .fjL^ B
i
«y /
y
//
X
/ / v%/
^/
S SS;
*
— ^^^
= ss: ^^^
B (loq let
Fig. 2 — Curves for the determination of /3„
Fig. 3— Relationship between F, and H
SHRINKAGE STRESSES IN CONCRETE
197
B (log scale)
Fig. 4 — Relationship between cos 0„ and B
B (log scale)
Fig. 5 — Relationship between H n and B
ato 
0.09 
0.08 
ao? 
o.ot> 
n
0.05 
0.04 
0.03 
o.oz
B ( log scale
Fig. 6 — Relationship between G n and B
198
JOURNAL OF THE AMERICAN CONCRETE INSTITUTE January 1946
T T~
f A T
IT \ $ .
P > I"
\ \~ \\_ '
\K\ \
^^^$\ V
l^^^^^sA
1 S? S 2 S g $ g S 2S°
t *
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o o
SI 8
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lit
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0
SHRINKAGE STRESSES IN CONCRETE
199
X
<* £ 8
<S^ 1 8
^? II
it*
c
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= .2
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a* §?
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L^. — #/ r
\\\ Z7Z^ C
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200
JOURNAL OF THE AMERICAN CONCRETE INSTITUTE January 1946
5*
.
Fig. 14Maximum stress (maximum value of stress at exposed surface) and
maximum warping vs. the parameter B
TABLE 10„
n tan n = B
B
0.0
0.1
0.5
1.0
2.0
5.0
10.0
100.0
0.311053
0.653271
0.860334
1.076874
1.31384
1.42886
1.55525
7T 2
3.1731
3.2923
3.4250
3.6436
4.0338
4.3058
4.6656
3tt 2
ft
2tt
6.2991
6.3615
6.4372
6.5783
6.9097
7.2281
7.7760
5tt/2
0%
3tt
4,
5w
9.4354
12.574
15.715
9.4774
12.606
15.740
9.5292
12.645
15.771
9.6296
12.722
15.834
9.8927
12.935
16.010
10.200
13.213
16.260
10.887
13.998
17.109
7tt 2
9*72
llir/2
_^
N CONCRETE
201
TABLE 2— RATIO OF SHRINKAGE (OR SWELLING) TO ULTIMATE SHRINKAGE
(OR SWELLING)
/
'
I
•
'
I JOURNAL OF THE AMERICAN CONCRETE INSTITUTE January 1946
TABLE 3UNIT SHORTENINGAVERAGE UNIT SHRINKAGE (= )
1 v ~ T ^u
= 1  2 e H„
0.1
B = 0.5
B = 1.0
2.0
5.0
B = 10.0
B =
0.001
0.002
0.003
0.005
0.010
0.015
0.02
0.03
0.04
0.05
0.075
0.10
0.15
0.20
0.30
0.4(1
0.50
0.75
1.0
1.5
2.0
3.0
4.0
5.0
7.5
10.0
15.0
20.0
O IIOIIl
0.0002
0.0003
0.0005
O OIIK)
0.0015
0.0020
0.0030
0.0039
0.0049
0.0074
0.0098
0.0146
0.0193
0.0288
0.0382
0.0474
. 0702
0.0924
0.1353
0.1761
0.2521
0.3211
0.3837
O ..161
0.6301
0.7658
B556
0.0005
0.0010
0.0015
0.0024
0.0048
0.0072
00U5
0.0141
0.0185
0.0231
0.0340
O 0446
0.0653
0.0854
0.1239
0.1606
n 1057
0.2771
0.3502
0.4751
0.5767
0.7233
0.8182
0.8821
0.0594
0.9860
i) 9983
i» gggg
O 0010
0.0019
0.0029
0.0047
o 009:1
0.0137
0.0181
0.0265
0.0347
0.0426
0.0620
0.0803
0.1154
0.1489
0.2064
. 2666
0.3188
0.4339
0.5296
0.6751
o 7756
0.8930
(I 04 SM
. 9756
0062
. 9994
1 oooo
0.0019
0.0037
0.0055
0.0090
0.0173
0.0252
0.0317
0.0473
0.0611
0.0742
. 1050
(I 1336
. 1860
0.2327
0.3190
0.3939
4604
II 5062
O 0970
. 8306
. 9052
O 0625
O 0907
0.9971
0.9998
1 . 0000
0.0045
0.0085
0.0124
0.0196
0.0370
0.0515
. 0649
0.0891
0.1115
0.1319
0.1779
0.2186
(i 2895
0.3510
0.4555
0.5422
0.6148
0.7498
0.8375
0.9315
0.9711
. 9949
0091
it 9996
1.0000
0.0080
0.0147
0.0206
0.0323
0.0556
. 0755
. 0932
0.1242
0.1512
0.1753
. 2282
0.2739
0.3510
0.4167
0.5258
0.6136
. 6849
0.8109
0.8865
0.9591
0.9857
0.9981
0.9998
1.0000
0.0357
0506
0.0619
0.0800
0.1129
. 1383
0.1596
0.1954
0.2257
0.2523
0.3090
3506
. 4370
504 1
0.6133
0.6979
0.7641
0.8726
0.9313
0.9800
0.9942
0.9995
1.0000
TABLE 4 WARPING (%~ in thousandths)
0.0010
0.0015
0.0020
0.0030
0040
0.0050
0.0075
0.010
0.015
n 020
0.030
(HO
0.050
0.075
0.10
0.15
0.20
0.30
0.40
o 50
0.75
1.0
1.5
2.0
3.0
4.0
5.0
7.5
10.0
15.0
20.0
B = 0.1
(I 04S
0.071
0.093
0.137
0.180
0.222
0.324
0.421
0.606
0.778
1.092
1.375
1.628
2.161
2.568
3.126
3.457
3.757
3,843
3.851
3 782
3.694
3.519
3.353
3.044
2.763
2 50 s
1.969
1 506
0.953
26 IW
2
1
T$\
B = 0.5 B = 1.0
2.0
B = 5.0
0.235
0.348
(I 15s
.74
883
1 (Ills
1 579
2 04 5
2.920
3.735
5.415
6.518
7 s:i2
10.138
1 1 . S46
14.017
15.128
15 773
15 546
15.043
L3 588
12 217
o 869
7 059
5.203
3.417
2.216
0.763
0.262
0.031
004
0.46
0.68
0.90
1.32
1.72
2.12
3.06
3.94
7.08
9.74
12 05
14.03
17.96
20 75
23.96
25 29
i:* 61
24.14
22.59
is S5
15.67
10.82
7 17
3 . 57
1.70
0.81
. 1 3
02
0.00
0.91
1.33
1.73
2.54
3.31
4.03
5.74
7.35
10.25
12 92
17.34
21.07
24.21
30 os
33.87
37.56
38.32
36.28
32.88
29 43
22.07
16.52
9.25
5 is
2.05
0.51
0.16
0.01
0.00
2.13
3 os
3.99
5.73
7.23
8.69
13 Os
1 5 . 1 s
20 4 5
24.97
32.63
38.32
42.70
49.98
53.81
55.70
53.89
47.03
39.90
33.64
21.86
14.20
5.09
2.53
0.45
0.08
0.01
0.00
3.82
5.43
S.89
9.61
12.03
14.27
19.16
23.35
30.38
35.92
44.71
50.93
55.33
62.22
65.16
64.50
60.56
50.52
41 .38
33 77
20.27
12.17
4.38
1.58
0.20
0.03
0.00
16.83
19.79
23 . 22
27.90
31.66
34.89
41.36
46 . 42
54.12
67.67
72.80
76.15
79.76
80.40
74.18
66.85
52.74
41.27
32.25
17.40
9.39
2.74
0.80
0.07
0.01
0.00
,
SHRINKAGE STRESSES IN CONCRETE
203
TABLE 5— RATIO OF STRESS a" T IN AN UNRESTRAINED BEAM DRYING FROM
TWO OPPOSITE SIDES (OR IN A BEAM DRYING FROM ONLY ONE SIDE
AND RESTRAINED AGAINST WARPING) TO THE ULTIMATE STRESS
FOR COMPLETE RESTRAINT ES*>
S
Sa
Si
v
y
y
y
V
y
y
V
V
y
v
V
T
=0
=0.2
=0.4
=0.6
=0.8
 = 1.0
=0
=0.2
=0.4
=0.6
=0.8
=1.0
b
b
b
b
6
6
&
b
6
b
6
b
B=0.1
B =
1.0
0.0
0.005
.0005
.0005
.0005
.0005
.0003
.0047
.0047
.0047
.0040
.0703
0.010
.0010
.0010
.0010
.0010
.(1001
.0093
.0093
.0093
.0091
.0000
.0942
.0O15
OKI 5
.0014
,0006
.0120
.0137
.0137
.0125
.1101
0.020
.0020
.0020
.0020
.0017
.0013
.0138
.0181
.0148
1 24
.1235
0.030
.0030
.(11)20
0019
2
.0265
.020 1
.0170
.0246
.1425
0.(11(1
.0030
.0030
.0019
.0041
.01 s.3
.0346
.0344
; 1
.0165
,1563
0.060
.0049
.00 is
.0019
.0051
.0198
.0123
nil',
.0361
.0152
<<l I 1
.1670
0.075
.0072
.0000
.001 »
.0072
.0228
.0570
.0112
.0104
0616
.1851
0.10
.0090
.OOSl
.1.110 1
.Hi .on
.0682
.0054
.1961
0.15
.0117
.0106
.0069
.oooo
.011 1
.0277
..010
.0821
.001 1
.0871
.2064
0.20
.0133
 0119
.0074
.0005
.0293
oooo
.0880
.0539
.0929
.2081
0.30
.0149
.0132
.0079
.0010
.013S
.0306
.1028
.0900
.0530
,0089
.0948
.2027
0.40
.0154
.0136
.OOS]
.0011
.0141
,0308
.0855
.0497
01)0 1
.1893
i >..,()
.0154
.0080
0013
.0142
.0311
.001 1
.0S01
.0088
.1766
0.75
.0152
.0133
.0079
.0013
.0302
.0764
.0669
.0074
.0699
.1471
1.0
.0148
.0130
.0(i77
.0012
.0295
.0634
.0555
.0321
.0062
.0582
.1222
1.5
.0140
.0123
.0072
.0011
.0131
,0281
.0439
.0384
.0223
.0401
.0844
2.0
.0134
.0118
.0070
.0011
.0124
.0267
.0266
.0153
.0029
.0277
.0583
3.0
.0 122
.0107
.0003
.0010
.0113
.0243
.0145
.0127
.0074
0013
.0131
.0277
4.0
.0111
.0007
.0057
.0102
.0221
.0069
.0060
.0035
,0007
.0133
5.0
.0101
.on.v.t
.0052
9
0092
.0200
.0032
.0028
.0016
.0001
.0030
(.004
7.5
.0070
.0069
noil
0006
.0073
.0157
ion
.006]
.0055
.0033
.0055
.0120
15.0
.0038
.0033
.0020
.0035
,0076
20.0
.0023
.0021
.0012
.0002
.0022
.0047
B =
5.0
£ = «>
0.0
0.001
.00 1".
.00 15
.0045
.0045
.1514
.1.357
.0357
.0357
.0357
.9743
0.002
.OOS5
.0OS5
.0085
.0085
.0084
.2010
.0506
.0506
.9494
0.0(15
.0100
.0196
.0100
.0124
.2823
.0800
.0800
.0345
.9200
0.010
.0370
.0370
.0370
.0020
.1129
. 1 1 20
 L127
.0412
.8871
0.015
.0515
.0515
.1383
.1383
.1378
.1171
.1100
.8617
0.020
.0649
.0646
.0523
.4120
.1596
.1595
.1141
.1577
3404
0.030
.0890
.Os.sl
.Osls
.0538
.0871
.4441
.1944
.8046
0.040
.1 100
.1098
.0998
.0481
.117.,
.1000
.2249
.2210
,10 1s
.0684
.2538
.7713
0.050
.12! tO
.1277
.1001
 0401
.1404
.4707
.2492
.2408
.1945
.0464
.2748
.7477
0.075
.17(10
.1604
.1207
.0200
.1765
.4778
.2892
.2683
.0073
0.10
 1965
.1700
. L230
.0044
.1948
.4727
.3096
.1771
.0151
it. 15
 2160
.1969
.llso
.0130
.2052
t hi
.3012
.0317
1 ». _'( 1
.2164
.ISO 7
.1112
.0199
.2765
.2404
.4959
0.30
.lols
.H.71
.0948
.0216
.17 10
.3551
.2200
.1000
.1048
.0292
.1986
.3867
0.40
.1634
 1 1J'
.OSOI
.OI01
.1485
.1723
.1492
.0818
.0231
,3021
0.50
.1370
.1 103
.01,7 1
0103
.1347
.1166
.0640
.0180
.2359
(J. 7.'.
 osoo
.0770
.0437
0106
.0727
.0629
.0655
1.0
.0582
.0506
.Ojsj
.1064
.0393
.0341
.01S7
.0353
.0687
.0246
.0214
 0170
.0223
.0448
.0114
.0100
.0015
.0200
2^0
.0104
.0051
. (HII2
.0094
.0034
.0029
.0016
.0004
.0030
.0058
.0019
.0016
.0009
.0016
204 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE January 1946
TABLE 6— RATIO OF STRESS <r, IN AN UNRESTRAINED BEAM DRYING FPniu
ONLY ONE SIDE TO ULTIMATE STRESS FOR COMPLETE RESTRAINT g?
0.0
0.005
0.010
0.015
0.10
0.30
1.5
0.001

f= + (« 12*)
Ox s
2bv.
v v y
=0.2 =0.4 =0.6
b b b
v y
= 1.0
6 6
B = 0. 1
y „ y y y y y
=0 =0.2 =0.4 =0.6 =0.8  = 1.0
b b b b b b
.0021
.0071
.0003
.0005
9
.0011
.0011
.0003
.0003
.0001
.0001
.0001
.0000

.0019
28
.0017
.0014
.0014
.0005
.0016
.0015

.0001
.0003
.0001
.0001
78
.0100
n


.0060
.0030

B
1.0
0080
.0198
.0343
.o;uo
.0419
.0510
.0473
0441
.0210
.0145
.0029
.0049
.0074
.0087
.VO\'0
.0089
.0042
.0030
.0022
.0014

.0010
.0001
.0022
.0046
.0070
.0091
.0135
.0169
.0193
.0226
.0237
.0243
22
 . '
.0102
.0160
.0133


.0031
.0015
.0006
.0072
.0138
.0192
.0233
.0310
.0320
.0320
.0303
.0274
.024 7
.0218
.0199
.0183
.0132
.0120
.0061
.0030
.0013
.0006
.0116
.0142
.0142
.0131
.0106
.0081
.0061
.0031
.0012
.0008
.0019
.0026
.0025
.0020
.001b
.0011
.0002
.0002
.0001
.0576
.0706
.0766
.0811
.0641
.0840
.0773
.0716
.0626
.0490
.0445
.0318
.0340
.0282
.0195
.0135
.0063
.0031
.0015
B = 5.0
.0141

.0011

.0001
.0000

5

.0004


28$
3
27
7
.0000
"
.1142
72
.0006
B =
887

728
>8S
.0014
.0411
.0105
.0027
.0001
.0009
.0006
.0002
.0002
.0000
.0000
.1044
.1031
.0641
.0416
.0021
,0006
 781
.121b
.1626
.1*23
.1741
78
.1031
.0814
57
.0459
.0340
.0111
.0018
.0006
.0963
.1326
.1601
.00b3
.0005
.0093
.0117
.0119
.0112
.006S»
.0053
.0029
.0015
.0004
.0001
.8101
.7107
.5370
.4817
.3375
.2905
.2122
.1670
.1179
.0545
.0010
Title 428 — a part of PROCEEDINGS, AMERICAN CONCRETE INSTITUTE Vol. 42
JOURNAL
of the
AMERICAN CONCRETE INSTITUTE
(copyrighted)
Vol. 17 No. 4 7400 SECOND BOULEVARD, DETROIT 2, MICHIGAN February 1946
Shrinkage Stresses in Concrete*
By GERALD PICKETTf
Member American Concrete Institute
PART 2APPLICATION OF THE THEORY PRESENTED IN PART 1
TO EXPERIMENTAL RESULTS
Carlson's results on prisms drying from one end
As mentioned in Part 1, Carlson 1 applied diffusion principles to the
problem of computing both loss of moisture and distribution of shrinkage.
The fundamental equations on which his computations were based are
the equations to which Equations 5 and 22 of Part 1 reduce when the
parameter B is set equal to infinity. In his experimental work the
prisms were allowed to dry through one end only, the rest of the surface
being sealed. Measurements were made over gage lines that were
parallel to the direction of flow of moisture. These conditions appear to
be most favorable for the direct measurement of the distribution of
shrinkage tendency since in an unrestrained specimen shrinkage stresses
should not have any appreciable effect on the unit shortening in the
direction of moisture flow.
In Fig. 3 of his paper Carlson showed two diagrams. One diagram
gave the distribution of shrinkage as measured after a definite period of
drying and the other gave the computed "distribution of drying" (loss of
moisture) for different assumed coefficients of diffusion for the m
period of drying. The observed distribution of shrinkage and the com
puted "distribution of drying" are in good agreement when the proper
coefficient is selected. However, as shown by Fig. 1 of his paper, the
measured loss in weight was not in very good agreement with the theory.
Carlson could have obtained slightly better agreement between theory
and measured shrinkage if he had taken surface conditions into account,
♦Part 1 of this paper was published in the ACI Journal, January. 1946, and includes (p. 194) the com
Pl VPr;5e^or e o f f Applied Mechanics. Kansas State College, Manhattan, Kan., formerly Portland Cement
ition Research Laboratory, Chicago.
(361)
362 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE February 1946
rS  ^>
■    .«  <y
_\  f5 _— r _r._ : m =zaoox/a1
f=a07/n./cs  oerd&y
 £ *r  cemeni
stfiaa  _■  f
Fig. 1 5— Comporiscn of observed and calculated course of shrinkage
i.e., used a finite value for the parameter B. However, had he done so,
the discrepancy between theory and measured loss in weight would have
been greater than that shown.
Carlson's work is important evidence in support of the hypothesis
that shrinkage of concrete approximately follows the laws of diffusion.
Shrinkage of prisms of various sizes drying from one or more sides
In the work done in this laboratory measurements were made on
gage lengths transverse to the direction of moisture flow. Since variations
in shrinkage along the path of moisture flow result in stresses transverse
to the direction of flow, the measurements include the strains produced
h J These however, the specimens are long compared to their
dimension m the direction of moisture flow and the stressstrain relation
is linear, then as shown in Part I the shortening of the central axi< will
be equal to the average shrinkage. The results to be discussed provide
a test of the theory for conditions in which both size of specimen and
number i j es are variable.
the unit shortening versus davs of drving for three
different sizes of prism* of the same mix and for the three different
drying conditions discussed in Part 1. .Mix A and cement M, described
in the Appendix, were used. The specimens were cured seven davs
under water. Each point is the average of the results from two pn~
lne curves were constructed from computations based on the theoretical
equations developed in Part 1. These equations, which give the theoreti
SHRINKAGE STRESSES IN CONCRETE
363
unit shortening = S a
cal relationship between unit shortening, the constants of the material,
and dimensions of the specimen have the form
\ k ¥ bf
where S a> is ultimate shrinkage for the assumed final drying, / is the
surface factor, k is the diffusivity factor, and b and c are dimensions
of the specimen. The exact form of the function, especially the way in
which c/b enters into it, differs with the drying condition.
The three constants £«,, /, and k were evaluated from average ex
perimental values for the pair of prisms of 2in. square crosssection,
drying from four exposed sides. From these same constants the curves
were constructed, as shown in Fig. 15, not only for this pair but also for
the theoretical unit shortening of the other eight pairs of prisms. The
agreement between the experimental values and the calculated curves is
fairly satisfactory except for two pairs of 3x3in. specimens, which were
observed to have cracked during drying and therefore could not be
expected to shorten in accordance with the theory.
Discussion of the validity of the theory on the basis of the foregoing data
The data from those specimens that did not crack, together with
the data given by Carlson, might seem to indicate rather conclusively that
shrinkage does take place in accordance with the theory develops I in
Part 1. However, such a conclusion would not be justified. A good fit
between an equation and experimental data is necessary but it is not
sufficient proof of a theory. Although constants in the equations of
Part 1 may be chosen so that the theory given there will be in good agree
ment with experiment for certain measurements on spc.imens under a
few different conditions, the theory should be expected to fail under
some other conditions since it rests on some assumptions that are not
wholly correct. Shrinkage is not linearly related to change in moisture
content; the flow of moisture in concrete does not follow the law of
diffusion; and the stressstrain relation is not linear. Since the assump
tions are not wholly correct, the factors S «> , /, k, and E that are supposed
to characterize the material must be empirical, and experimentally
determined numerical values of these factors will be different for different
Tests on the same material. The good agreement between theory and
the experimentally determined contraction of the specimens discussed
above must be the result of the balancing of opposing effects. They u ill
not necessarily balance the same way in another test.
The foregoing criticism means that however promising the theory
may appear from the results of a few experiments, the application of the
theorv must be limited and extrapolation of the results to sizes of speci
mens or conditions of drying other than those for which the constants
364 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE February 1946
were determined cannot be made with confidence. The selection of
•r given conditions constitutes the chief difficulty for the
prad of the equations. This doee not mean that the theory is
of little I considerable value.
Tnn l equations are not rigorously correct and the
*ants cai the meaning attached to them. But the
linkage does follow the diffusion equation approxi
mately and that the defn a are approximately those
gi the theoretical equations if the empirical constants selected
to give fair agreement with experimental result.
Although the experimental results shown in Fig. 15 appear to be
in good agreement with the theory, a close study shows the following in
regard to those pri~ og from only one side: I 1 After pro
i of the prisms drying from one fflde I
 • ^ indicated by the theoretical curv<
 :u. prisms deviate more from
•' Lxlin The lxlin.
the same lvalues as the 2x2in.
till more I
   tion. I
Lid result in still greater di>
omputed values unless allow]
th change in lvalue.
Comparison of dota on warping with data on shortening
ded by the results shown in
•.re root eriod
! the prism.* As mentioned in
is The abscises give* a nearly straight
i entering and warping.
■ the thickness of the specimen puts
able basis. Multiplying the
 span by the thickness put* the
on the same basis in regard to unit defoi
J j l
 15 that were drying from only

trying fn

OS.
iiammc* in the dmiio: kniu
SHRINKAGE STRESSES IN CONCRETE
365
4000
V)
JZ
t—
c
o
V)00
b
c
en
c
2000
c
0)
I—
o
x:
1000
»
c
3
<^
in
.40
<U
c
r
&
S.30
r\,
v>
*)
<L>
E
t)
t—
£.20
c
h
o
<ij
^
<
u
tj
dj
.C:
Q)
*.10
o
Ci
t—
a
OJ
0
h
f
QJ
*
CJ
Points are Experimental I
Curves are Theoretical, based on
/*  o.o7in. per day
— k a 0J5 /a ? p £T flto^"
12
10
8
• Experimental for 2 in Prism
o « " //A
I [^
 JoAC curves \f
i u
o. 07 in. per day
■ 0.035 in. 2 • "
f^oo 2ff00*/0~ tf
Dashed curves if *o. 032 in per day
\k * o.o zo in. 2 u
Z 4 6 8 10 12
Square Root of Dans Drying _ Vt~
Thickness fn inches b
Fig. 16 Comparison of observed and calculated course in shortening and warping of
prisms of mix A and cement M
The curves in the upper diagram and the solid d the i
diagram were constructed from the theoretical equations using the
same values of the constants S », /, and k as were used in construe
the curves of Fig. 15. Therefore, the three curves in the upper diau
of Kg. 16 represent the same equations as three of the curves in Fig. 15;
they differ only in the abscissas. The solid curves in the lo jram
deviate considerably from the plotted point. This indicate
greater disagreement with theory than is shown in Kg. 15. II
the dashed curves obtained from the theoretical equation by using the
Bame valtie of Sea but with / reduced by 54 per cent and k reduced by
43 per cent are in very good agreement with the experimental values.
366 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE February 1946
The computed shrinkage stresses will be about the same whether the
first or the reduced values of/ and k are used.
The values of the constants £», /, and fc, used in constructing the
dashed curves, were obtained from three measurements as follows: (1)
maximum warping of the 1in. specimen, (2) time at which this maximum
warping occurred, and (3) final shortening of a companion specimen.
The agreement throughout the course of drying between the experi
mental values for warping and those given by the theoretical equations
when these constants are used is excellent for the 1in. specimen and very
good for the 2in. specimen.
The above shows that if data on warping and data on shortening
are analyzed separately, either group of data will appeal to be in accord
with the theory if the thicknesses of the specimens do not differ too much,
but the values of/ and k obtained from the two groups of data will be
different. The fact that the factor k is an empirical rather than a fun
damental property of the material is believed to be the chief reason
why both groups of di * nted satisfactorily by on<
nits. The empirical nature off is considered to be of only
e in this study because it has much less effect than
■time arid warpingvs.time relations.
Effect of differences in k on warping and shortening
diffusivity on the theoretical value of
ping (Equ 2 bortening (Equation 2:>> : are shown in Fig.
17 v ' plotted against the parameter V// jTfor
■ ■■■ fc. As shown, differences in k have
ply warping; each curve follows the same
maximum point. The lower k t the greater
m maximum value of warping, of
ticipated because of the effect of k through tb
As shown by the curves for
I upon shortening is entirely different
 The rate of shortening is materially reduced
n the maximum shortening is unaffecfc
i rent effects of changes in k on
irping has been useful in explaining differences in
made with cements of different composition.
I results from concretes made with two diff
  Mix C was used. A comparison of
7 e conclusion that the coefficient of di
nt No. 515001.9 IS loner than for con
SHRINKAGE STRESSES IN CONCRETE 367
cretes made with cement No. 115001.9*. Concretes made with cement
from clinker No. 5 shortened at a lower rate but according to data not
plotted eventually shortened more than concretes from clinker No. 1.
Before this explanation was found, it seemed surprising that of two
groups of specimens drying from one side only, subjected to the same
exposure, one group would warp more and shorten less than the other
group. In order that one specimen warp less than a second when the
two specimens have the same average shrinkage, the distribution of
shrinkage in the first specimen would have to be more nearly uniform.
For the same surface conditions, a large value of k through the para
meter/6 k, tends to make shrinkage more nearly uniform and therefore
is accompanied by less warping. An increase in uniformity of shrink
age also reduces the shrinkage stresses in an unrestrained specimen
and therefore reduces the tendency for spontaneous cracking. (Fig. 14
Part 1 — shows how the theoretical maximum stresses depend on the
parameter fb/k (= B) .)
Effect of alkali content on k and its possible effects on cracking
It had been observed from various laboratory tests designed to measure
cracking tendencies that concretes made with cements from clinker
No. 5 tended to crack more than those made with cement from clinker
No. 1, even though measurements often showed less volume change at
the end of a given period of drying for the concretes of clinker No. 5.
This greater cracking tendency of cement from clinker No. 5 was at
tributed to its higher alkali content, since this appeared to be the only
important difference in their chemical compositions. Attempts to
evaluate k for concretes made with cements from these two clinkers
showed that for the same mix proportions the value of k for concrete
made with the cement of higheralkali content was only onehalf that
made with the cement of loweralkali content. These observations sug
gested the possibility that: alkali reduced k, a reduced k resulted in
higher shrinkage stresses, and higher stresses resulted in more cracking.
To investigate this effect of differences in alkali content more fully,
several tests were made using cement No. 116652.48. The procedure
was to add 0.91 per cent Na 2 by weight of cement in the form of NaOH
to the mixing water of one of two companion mixes. The results of one
using mix B are shown in Fig. 19 where shortening and weight losses
of prisms are plotted against period of drying.
The dimensions of the prisms were 2^x23^x1134 in. They dried from
all surfaces except the ends. By using for the specimens containing added
alkali a timescale equal to onethird the scale used for the regular
specimens the corresponding curves for both sets of specimens approxi
*As explained in the Appendix, the first number is the clinker number, the second is the specific surface
 Wagnea mel hod), and the third is the percentage of S0 3 .
368 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE February 1946
Fig. 18— Warp
ing and shorten
ing of prisms that
differ primarily in
the alkali con
tent of th«
ment used
Prisms 3in. thick in
the direction of mois
ture travel. Deflec
tion measured over a
32in span. Mix C.
Cured 7 days.
Fig. 17 — Theoreti
cal effect of k on
the course of con
traction and warp
ing of prisms dry
ing from one side
only
4 6 8 10 12 I4
le
2 4 6 8
10
Square Root of Number of Days Drying
20 30 40 50 60
Period of Drying days (without added alkali)
6C 90 120 ISO I6C ?io
Penod of Drymgdays (alkali added)
Fig. 19— Shrink
age and weight
loss for speci
mens with and
without added
alkali
SHRINKAGE STRESSES IN CONCRETE 369
mately coincided, indicating that the main effect of the added alkali
was to reduce the diffusivities for both shrinkage and moisture flow to
onethird the value without added alkali.
The effect of added alkali in reducing the diffusivity of shrinkage
for cement of clinker No. 1 is in accord with data reported by Haeger
mann. 21 Haegermann was primarily interested in the effects on shrink
age of additions of various sulfates to cements of different C 3 A contents.
The sulfates tried were ferrous, calcium, magnesium, sodium, and po
tassium. The amounts added were such as to increase the S0 3 , content
1 per cent, based on the cement. Five cements ranging from 15 per cent
computed C Z A content to zero per cent C 3 A were investigated.
The data were presented by Haegermann in the form of curves. For
each cement, the curves representing the sodium and potassium sulfate
additions are of noticeably different shape from the other curves for the
same cement, the difference in shape being such as would result from a
lower diffusivity. Since Haegermann did not give data on loss in weight
during drying, it can only be inferred from the data on shrinkage that
the sodium and potassium sulfates also reduced the diffusivity of moisture
flow.
From theoretical consideration, it appears that any highly soluble
material should reduce the relative rate of drying; i.e., should increase
the time required to lose a given percentage of the total amount of
moisture to be lost.* However, since many other factors affect the
rate of shrinkage, and alkalies have many other effects which may in
directly affect shrinkage, one should expect many real and apparent
contradictions to the above indication that an increase in alkali content
will retard shrinkage.
The effect of the added alkali on cracking was investigated by means
of the "wedge test"t and by the " restrainedshrinkage test" (subse
quently described). The result was that specimens of higher alkali
content showed a much greater tendency to crack, as measured by these
tests.
Other tests made in this laboratory show that for cements containing
an appreciable percentage of tricalcium aluminate, an increase in alkali
content will increase final shrinkage of laboratory specimens unless the
increase in alkali is accompanied by an increase in gypsum. The greater
♦This reasoning is based on the supposition that at least part of the flow of water in concrete is by means
of the following cycle: evaporation at an airwater interface, vapor diffusion across air space, capillary flow
?n feJdfSEd Space and again evaporation at airwater interface. .Since the diffusivity of the soluble ma
terial within the liquid is finite rather than infinite, at any airwater interface at which water is evaporating
ISL of soluble material will be higher than that for equilibrium with the adjacent liquid and
therebv tend to restrict evaporation at this interface, and at any airwater interface at which water is con
densing the conrei.tration of soluble material will be lower than that for equilibrium with the adjacent
Houid and therebv ten<l to restrict condensation at this interface. Therefore any highly soluble material
should retard the'drving bv reducing the diffuivity of moisture flow. .„.,.,. „ ,
t A speninei! L cast 5f1 he form of a wedge and. after curing, is permitted to dry from the two nonparallel
surfaces.
370 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE February 1946
tendency to crack of the specimens with higher alkali content might
have been due, at least in part, to a decrease in diffusivity and an in
crease in final shrinkage.
On the other hand, the possible benefits from alkali should not be
overlooked. The lowered rate of moisture loss will permit the interior of
concrete to retain sufficient moisture for additional hydration for a longer
time after drying of the surface begins. Prevention of complete drying
of the interior during the usual drying season should be especially ad
vantageous in preventing cracking when restraints against shortening
are present. Tests in this laboratory have also shown that concretes
of higher alkali content have greater capacity for plastic flow, which is a
favorable property.
Decrease in k as drying proceeds
As drying proceeds, the value of the coefficient of shrinkage diffusivity
t apparently decreases. This decrease no doubt results from a pro
gressive decrease in the apparent diffusivity of moisture, diffusivity of
moisture probably being a function of the moisture content. If diffusi
vity of moisture is a function of moisture content, then the shrinkage
diffusivity can be considered to be a function of the shrinkage S and the
differential equation becomes nonlinear. Adding particular solutions,
as was done in Part 1, is then not permissible.
However, if in place of considering k to be a function of the dependent
variable S it is considered to be a function of the independent variable
t and of the dimensions of the body, then the differential equation re
mains linear. Furthermore, if the factor/is considered to varv with time
in a hke manner so that the ratio f/k remains constant (see, for example
lation 2a), then all of the equations for displacements, stresses and
'  developed in Part 1 still apply if the symbol t appearing in then,
d by a function of t and the dimensions. The changes suggested
•■'iH.ve amount to a , ontinual change in the timescale so that the time
squired for given conditions to develop becomes progressively longer
 ,n,,dl,vl "« the theory in this way better agreement with experimental
results can he obtained.
Fig. 20 is an example of applying the foregoing analysis. The plotted
points are from experimental data on the average warping of four 3in
specimens ol concrete of mix B with cement 12280]. 94. When an
attempt „ras made to select constant values of/, Ife, and .<J. to be used
"' "■ theoretical equation that would give curves in agreement w,.h all
•> the experimental values, no( all 1he (1:((a ( , )ljl(i 1)( , bK)ughl ^
men wnh the theoretical equation. But by taking the following
lor the factors, a Letter fit was obtained.
SHRINKAGE STRESSES IN CONCRETE
371
0.06
002
n nr\
a
2 3 4 S 6
Square Root of Dags Drying VT
Thickness in inches b
Fig. 20 Com
parison of theo
retical and ex
perimental warp
ing
Points are from the
average warping of
four 3  i n beams.
Span 32inches. Mix
B. Cement 12280
1.94. Cured 7 days.
ft = 0.10J^j^inVday
■fh
f = 1.67 ft in dav, i.e.,  = 5
k
Soo = 765 x 10' 6
When these values are introduced into the differential equations and a
solution made, the symbol T in the final equations for warping, etc., is
replaced by
±k
k [
2 + t
1
b 2 L^ 2
where k is the initial value of k or 0.10 sq. in. per day. For convenience in
making computations preliminary to plotting of the theoretical curve,
b 2 T b A T 2
t was expressed in terms of T, or t = 1 . The tabular values
used for constructing the curve are given below :
From Table 4
Computed Values
2bv max
<t
T
3M»
i
b
V max b
0.01
0.0152
1.00
0.333
0.0178
0.03
0.0326
3.61
0.635
0.0384
0.10
0.0538
li). 1
1.46
0.0634
0.15
0.0557
36.4
2.01
0.0655
0.20
0.0539
58.5
2.55
0.0635
0.30
0.0470
118
3.62
0.0553
0.50
0.0336
298
5.75
0.03
0.75
0.021U
639
8.41
0.0258
The better agreement that can be obtained by the modified theory
probably would not compensate for the extra work in all cases. Since the
372
JOURNAL OF THE AMERICAN CONCRETE INSTITUTE February 1946
Fig 22 Shrink
oge of slabs of
n e s s • s drying
from two oppo
des
Fig. 21 Com
parison of theo
retical and ex
perimental warp
ing of beam ab
sorbing moisture
from one side
Poinli MPMHl th«
warping ol a 1 in
b»om, having all but
on* of i»t *urfac*s
»»ol»d, during »ub
mtnion in water
Cured 7 da yi and
rh»n dn#d at bO
R H for 10 month •
b*for« Ivilmg Mi*
6 C»m»nl 19980
1 94 Span3Jinchei
'
.l..
the valm
Worp,ng ol prisms during absorption of moisture from one s.de
SHRINKAGE STRESSES IN CONCRETE 373
zero warp where it remained for the rest of the test, a period of one
month. The experimental value for the warp after 6 hours' exposure is
considered to be in error. Other tests on 1in. prisms did not show the
indicated large decrease in warp between the fourth and sixth hours of
exposure.
Although the amount of experimental data on swelling is yet small,
the indications are that the application of the theory as developed in
Part 1 is limited, first, because at the beginning of wetting the moisture
already present will ordinarily not be uniformly distributed; second,
because of having remained wet longer, the cement in the interior re
gions will have hydrated more than that closer to the drying surface;
and third, as the concrete becomes wet again, hydration again starts.
Agreement with diffusion theory is not expected while hydration is
occurring at an appreciable rate, especially if the formation of hydration
products causes expansion.
Effect of thickness on rate and amount of shrinkage of walls or slabs
In an investigation of the effect of wall or slab thickness on the rate
and amount of shrinkage the results shown in Fig. 22 were obtained.
The specimens from which the data were taken were made of mix B.
Cements of two different compositions and a fine and a coarse grind of
each are represented. The specimens were cured seven days under
water. The specimens were 34 inches long and of either 2x6 or 2xl2in.
cross section. By sealing all but two surfaces the prisms were made to
represent slabs or walls of 2, 6, and 12in. thicknesses drying from two
opposite sides. For example, the specimens that represented a wall 12 in.
thick were 2xl2x34in. and dried from only the 2x34in. surfaces, there
being 12 inches between these surfaces.
Sets of gagepoints were cast in these pseudo slabs so that the short
ening over three or four 30in, parallel gage lines could be measured on
each specimen. Details are shown in the Appendix. Each curve was
obtained by averaging the results from four specimens of a kind.
As shown in Fig. 22, the results from these concrete specimens are in
general similar to those obtained on the cementsilica mix discussed
previously (see upper diagram, Fig. 16.). The curves have the charac
teristic Sshape found for similar plotting of data from smaller specimens.
The thicker the slab the greater its fb/k ( = B) and, according to theory
as shown by Fig. 8 (of Part I), jthe greater the shortening should be for a
given value of the abscissa, ^t/b. The experimental data are partly in
agreement and partly in disagreement with the theory in this regard.
In the middle, straightline portions of the curves, the curves are in the
correct positions relative to each other, but in ever y case the relative
positions become reversed at larger values of Vf/6. Also, the relative
374 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE February 1946
presenting the two < rinds are rev<
 portion of the curves. This latter deviation
on the basis of a nonlinear stressflow relation
 the negative, the specimen will
>uld if plastic Bon did not take place. Of
and negative Bows are equal (algebraic
the specimen is not changed by plastic Bon
in an unrestrained Bpecimen if the
However, if the stressflo* relation
bhe range of stresses developed, the
than the first power of th<
o( tin specimen is reduced by pi
pronounced foi the thick<
I Jurve H of Fig i i I
the
'!..• basu of a aonlin<
ol the Brat p
rind* wei
„,i enti
rig the I
othei teste with
Plorf.c How
•
SHRINKAGE STRESSES IN CONCRETE 375
In the usual measurements of plastic flow the quantity measured is
the inelastic deformation of a body that results from applied loads.
From these measurements computations are made of the inelastic defor
mations of the individual elements, i.e., average unit deformation if
the load is axial or unit deformation of the outer fiber if the load pro
duces flexure. If stresses from other sources are not present, the com
puted values may be representative of the actual plastic flow. But if
stresses from other sources are present, the computed and actual values
may differ appreciably. Therefore, if in addition to load stresses a
specimen is under stress as a result of nonuniform temperature or non
uniform shrinkage, it should be made clear whether the term plastic
flow refers to the resultant plastic flows of elements or to only computed
plastic Bows produced by loads. Since the effects of load and the effects
of < Irving are not simply additive, there is no clear basis for deciding
how much of the total deformation is due to the stresses arising direct!}
from the load. In agreement with previous writers, the deformations
produced by loads will be taken as the difference between the deforma
tions of loaded specimens and the deformations of identical specimens
under the same drying conditions but not under load. Only the defor
mations produced by loads will be computed and n I ><\ i
but in the interpretation of results consideration will be given to what the
actual inelastic deformations are believed to
As shown by the formulas for plastic flow used in this paper, the t
deformation produced by load is divided into tv ~, elastic and
inelastic. The elastic part is considered to be that which would be
recovered immediately if the load were rein ed it is determined
the computed loadstresses and the "dynamic" modulus of elastii
The remaining part is considered to be the plastic How produced by the
load.
Some investigators make a slightly different division in that the
elastic deformation is considered to be that which was product
diately upon application of the load rather than that which would
recovered immediately upon removal of the load. The two \
equal if the modulus of elasticity does not change during the test. Some
writers prefer to divide the total deformation produced by load into I
parts: ( 1) that recovered immediately upon removal of load, (2) that not
immediately but eventually recovered, (3) the permanent deformation.
McIIcnrv ' restricts the use of the term plastic flow to the third part.
This division into three parts has merit, especially for those
which the second part is an appreciable percentage of the total. For the
data given in this paper no separation of the second and third parts
could be made, but the permanent deformation (3) is believed to be
much greater than the temporary (2).
376 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE February 1946
If the specimen is not shrinking while it is under load, then a con
siderable part of the inelastic deformation is probably only temporary
and apparently the result of viscous flow in the adsorbed water films.
After removal of the load, the elastic constituents of the gelstructure
tend to restore the original shape but are retarded by the viscosity of
the adsorbed water films. However, if an elemental volume* of cement
paste is shrinking while under stress, the conditions are different. The
loss of moisture introduces relatively large interparticle forces which
tend to change the relative positions of the colloidal particles within an
element. Some adjacent particles are pulled closer together but others
are moved further apart. During this time of movement the directions
of relative motion of the particles may be appreciably affected by stn
on the element. In this v ses on an element during the time it is
shrinking may produce comparatively large permanent deformations.
The foregoing is one explanation for the much larger amount of
plastic flow that a load will produce on a drying specimen compared to
what it would produce if either tin specimen were prevented from drying
or had pr< been dried. It is also an explanation of the relatively
• near the drying surface to deform plasti
cally without cracking. I) the analysis is correct, then a definite stn —
flow relationship cannot l»< given element of concrete since
vould depend not only upon the magnitude
duration i on an element but also upon the changes in mois
d while the element was under stress.
formation was considered to be per
i in deformation with time was con
hiefiy by changes in distribution of shrinl.
i ly small lag in time after the develop
!n "pla8tic Mow" rather than "creep"
;>.i! t of t he deformation.
hat way the patic flows of the individual
deformations ol the body as a whole.
dal load the plastic flow in tenion or
. the load is the difference in the a um
[nation ol each dement and what the algebraic
body bad not been under load. But the pis
ral load depends on the moment of the inela
■ lement with neutral axu " Both
aon side and plastic compression on the
.1 load contribute to the measured
'
SHRINKAGE STRESSES IN CONCRETE 377
Summary of remarks on plastic flow. The actual plastic deformation
of elemental volumes of a specimen may be much different from that
computed on the basis of laboratory experiments if shrinkagestresses
are present, but in this paper the plotted curves represent such computed
values. Computed values are based upon the difference in the deforma
tions of loaded and not loaded specimens. The term "plastic flow" is
used in this paper to refer to either actual or computed plastic deforma
tion. Plastic deformation is arbitrarily defined as that part of the total
deformation produced by stress (either by actual stress or by loadstress
as indicated by the text) that would not be immediately recovered upon
removal of the stress.
Effect of thickness on stresses and plastic Flow when the slab is partially restrained against
shortening
Companion specimens of the same size and sealed in the same manner
as those represented in Fig. 22 were partially restrained against shrink
age by specially designed steel bars, somewhat as were those described
by Carlson. 23 The main features of the steel bars are shown in Fig. 23.
(The concrete specimen illustrated in Fig. 23, however, is from another
test in which the concrete was allowed to dry from all sides and only one
bar was used per specimen). Each specimen of 2x6in. cross section con
tained two %m. diameter bars, and each specimen of 2xl2in. cross
section contained four 5^in. diameter bars. The arrangement of bars
is shown in the Appendix, and in Fig. 24.
A rubber tube covered the central 20 inches of each bar so as to prevent
bond over a 20in. gage length, thereby insuring the same axial force
in the bar over all sections of the gage length. That part of each steel
bar not covered with rubber was threaded and thus the bars were anchored
to the concrete for a distance of 7 in. on each side of the gage length. Be
cause of this anchorage the shortening of the steel bar over the gage
length is equal to the shortening of the concrete over the same gage
length. Moreover, as is obvious from considerations of equilibrium,
the force in the concrete in this gage length is equal and opposite to the
force in the steel in the same gage length. Therefore, the average unit
stress in the concrete can be computed from the change in length, modu
lus of elasticity, and percentage of steel. The formula is
A S E S M
(Jr =
A c I
where <x c is average stress in the concrete
A 9 is crosssectional area of the steel
A c is crosssectional area of the concrete
E 8 is Young's modulus for the steel
378 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE February 1946
n (
dUb
vHHIHHK
• [
fS
c
01
E
u
V
a
(/)
r
c
N
CO
5 1
i
W
\
\ v 1
^r
0 9 — 9*T
_ * 8 ° S
— Mr i
SP«Z P/,;#
■ V/ £ 2Qn± J*qqn# yjoouj?
i =
i\
o
*■_ «t >
I
<3
■ it
c . .'C
■z iz 9
SHRINKAGE STRESSES IN CONCRETE 379
A/ is net change in length over gage length after corrections
have been made for any change in temperature.* id is nega
tive if the specimen has shortened.
/ is gage length.
Not only the average stress caused by the restraint but also plastic
flow caused by this stress can be computed if the modulus of elasticity
of the concrete is known and the assumption is made that the shrinkage
tendencies of the restrained specimens are the same as those for com
panion unrestrained specimens of the same size. The formula is
° = S " B. + 1
where c is unit plastic flow caused by restraint,
S av is the unit shortening of the freeshrinkage specimens, and
E c is Young's modulus for the concrete.
Performance of partially restrained specimens. As explained in the
Appendix all specimens were cured under water. The specimens tended
to expand during this storage and consequently the concrete in those
partially restrained with restraining bars was compressed. Therefore,
for a short time after drying began, the direction of the plastic flow
produced by the restraining bars was in a negative direction. Shortly
after drying began, the stresses in the restraining bars changed from
tensile to compressive, and the average stress in the concrete changed
from compressive to tensile.
Under the conditions of this test the average stress reaches a maximum
and then slowly decreases if failure by spontaneous cracking does not
occur. A specimen's average stress and its shortening necessarily reach
their maximums simultaneously if the temperature remains constant.
Therefore, the time of maximum average stress is the time when the
rate of average shrinkage equals the rate of plastic deformation. During
the decrease of average stress, the rate of plastic deformation exceeds
the rate of shrinking.
Ordinarily in this test the specimens are not permitted to reach a final
equilibrium state in regard to shrinkage, shrinkagestress, and plastic
flow. But just after the maximum restraining force has been developed
additional tensile load sufficient to cause failure of the specimen is
applied. This load is applied to the protruding threaded ends of the
restraining bars by a machine designed for the purpose. While the load
is being applied, measurements are taken so that the added stress in
the concrete can be determined. Further details are given in Fig. 33
of the Appendix.
♦All tests were conducted in a room maintained at 76 =±= 1°F and a relative humidity of 50 =•= 2%,
except for occasional deviations from these limits. #
380 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE February 1946
500, , m
Fig. 25— Plastic
flow in the par
Hall/ restrained
specimens of Fig.
24
Cement 12260 /. 94
Factor of safety. The purpose of the testing just described is to
learn how close the specimen comes to cracking spontaneously The
ratio of the computed stress at failure to the maximum average shrinkage
stress is called a factor of safety. Specimens that crack spontaneously
are reported as having a factor of safety Less than unitv. Results showing
computed average shrinkagestresses and plastic deformation are shown
in Fig 24 and 25, respectively. The record of the number of specimens
of each cement that cracked spontaneously and of the average factors of
safety (f.s.) of those that did not crack is also shown in Fig. 24.
Effect of thickness on plastic flow. Attention is called to the similarity
ol the three sets of curves in Fig. 22, 24, and 25. The similarity is not
to be interpreted warily indicating that the plastic deformation
is proportional to its stress. One is tempted to make this
1 " t " , , , 1 "V e » « it were true, then the plastic flow of a specimen
would depend only on average si ress and not on the distribution of stress
In general, the diagrams show that the 12in. specimens developed con
siderably more plastic flow for the same amount of average stress than
«<*  or the 6in. specimens.
T1 >i possible reasons fortius: first, because of the lower
temng, the larger specimens will have been under a given
 longer than the smaller specimens and therefore would be
expected to have more plastic flow for the same stress. Since the time
^quired for the same amount of shortening is approximately pro,
" i:i <" "'<• 7—  the thickness, the 12in. specimens will in genial
have been under a given range of stress about four times as long as the
lH ": ~»™ Second, sin,,, the thicker specimens will have higher
maxnnum etresses, the additional plastic flow could be accounted for
by ■ nonhnear stressflow relationship whether or not this reTtionsh*
SHRINKAGE STRESSES IN CONCRETE
381
for each element was modified while the element was losing moisture
rapidly. Third, the assumption that shrinkage tendencies of the re
strainedshrinkage and freeshrinkage specimens are equal is not entirely
correct and consequently their computed plastic flow are in error. A
difference in the shrinkage tendencies of the 12in. free and restrained
specimens might result since the arrangement of the four bars was such as
partially to obstruct the flow of moisture.
Probably all factors listed above contributed to the results. Of the
factors causing the computed plastic flow to be greater in the 12in.
specimens, the author is of the opinion that the nonlinear stressflow
relation contributed much more than the difference in duration of given
stresses.* That the maximum stresses in the larger specimens are
higher is shown by the fact that all the 12in. restrained specimens of
three of the four cements cracked (see Fig. 24) whereas only a few of the
6in. specimens and none of the 2in. specimens cracked.
According to most of these arguments the 2in. specimens should
have less plastic flow than the 6in. specimens, whereas in general they
have slightly more for the same shortening and for the same average
stress. A complete explanation for this is not at hand, but the lesser
extent of hydration of the cement in the 2in. specimens because of their
more rapid drying may be a factor. Also, the exposed surfaces of the
2in. specimens were the top and bottom surfaces as cast, whereas the
drying surfaces for all the other specimens were the sides as cast. Bleed
ing and settlement of the plastic mix before initial hardening
plete always makes the concrete near the top and that near the bottom
as cast different from that at the sides. It must also be remembered
that the computed plastic deformation may be more or less than the real
plastic deformation of the material.
Use of beams drying from only one side for determining probable stresses in slabs or walls
drying from two opposite sides
To obtain information on plastic flow and on the magnitude and
distribution of stresses in unrestrained walls or slabs drying from two
opposite sides another set of specimens, also companion to those repre
sented in Fig. 22, were made. These specimens differed from those of
Fig. 22 in that they were permitted to dry from only one side in
of two opposite sides and in that the thicknesses of corresponding speci
mens were just half those of Fig. 22. Since they were half as thick and
dried from only one side instead of two sides (see Appendix), any one of
se specimens was considered to have the same conditions of di
and consequently the same distribution of shrinkage tendency as either
half of a corresponding specimen represented in Fig. 22.
♦Most contemporary writers on the inelastic properties of concrete apparently would take the opposite
view. ThS difference "in viewpoint » explained and an argument for the author's view is given in Ref. 20.
382 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE February 1946
If the distribution of shrinkage tendency is the same and if the correct
external forces are applied so as to make all the deformations the same
as those of either half of the corresponding specimen, then the distribu
tion of stresses will also be the same.
Eight specimens of a kind were made, four of which were allowed
to warp freely and four were restrained against warping. It was not
feasible to distribute the external restraining forces on the specimens
to be restrained against warping in exactly the manner that the mutual
forces between the two halves of the corresponding specimens were
ibuted. As shown in the Appendix, the method adopted ws
support the specimen as a simple beam and to apply enough force at the
quarterpoints to prevent warping of the central half.
Ifi discussed previously and as shown by Equation 19*, the amount
of warping of a specimen free to warp is indicative of the nonuniformity
linkage tendency. From similar considerations it follows that the
sary to prevent warping is indicative of the non
uniformil ee resulting from the nonuniformity of shrinkage
Furtli. the difference between the actual moment
required ing and that computed from the amount of
flow, gives an indication of the dis
tribution f ] one of the four cements are
give the actual moment developed,
livided I ection modulus, / c, for beams of three different
what the Mc I would have been
ment had not produced plastic flow. The upper
measured values of the warping tendency
and x ; panioo srx cimens. Foung's modulus was
1 quency of vibration.
^ ace the ordinatee in Pig. 26 are
i at the stress in the outer fiber according
Formula. According to the lower curves of
or the 1 in. specimens reached B maximum
kefirs* day ol drying. Thecomputed
3in specimen was 190 lb. per sq. in. and
mputed Btreas in the 6in. specimens had
and the indications are that, had the
uued, the computed beds maximum
• r about LOO days ol drying. om
reon the!
The acta ■ the drj
rapidly
SHRINKAGE STRESSES IN CONCRETE
383
Fig. 26 — Com
parison of actual
moment neces
sary to keep
beams from
warping with the
conputed mo
ment necessary
to strai g hten
companion
beams that are
free to warp.
(Plotted in terms
will build up as rapidly in the thicker slabs as in the thinner. The
actual stresses are probably better represented by the solid curves of
Kg. 27.
Stresses based on modified theory. The solid curves of Fig. 27 show
stresses based upon a modified theory. These curves were obtained by
substituting appropriate values of the parameters y/b, kt/b 2 ,fb/k, E, and
So into Equation 20 of Part 1. For the construction of these curves the
theory as presented in Part 1 was modified in that, instead of using
constant values for the factors k, f, Soo } and E, the following procedure
was pursued:
(1) The ultimate shrinkage S*> was set equal to 750, 700, and 600
millionths, respectively, for the 1, 3, and 6in. thick specimens. The
selection of these separate values rather than one value for all specimens
was governed by the apparent ultimate unit shortenings of the cor
responding freeshrinkage specimens (Fig. 22).
(2) The ratio f/k was set equal to 2.5 in." 1 , i.e., fb/k was 2.5 for 1in.,
7.5 for 3in. and 15.0 for 6in. specimens. When this value of f/k and
the above values of &«, were used, the theoretical maximum values of
warping as given by curve A of Fig. 14* were found to be in agreement
with the experimental values of maximum warping for each of the three
thicknesses of specimens.
(3) A value of kt/b 2 was selected for each period of drying (1, 7, and
28 days) and for each thickness of specimen, such that when substituted
along with the above values of Sa> and f/k in Equation 23* for warping
the result would be in agreement with the experimentally determined
values for these periods and these thicknesses.
(4) A value of E was selected for each period of drying and each thick
ness of specimen such that the theoretical moment given by Equation
27 would be in agreement with the experimentally determined values.
♦See Part 1.
384
Fig. 27 Theo
retical distribu
tion of shrinkage
stresses (modified
theory) in slabs
drying from two
opposite surfaces
for various dry
ing periods and
slab thicknesses
JOURNAL OF THE AMERICAN CONCRETE INSTITUTE
12 in. Slab 6in.5!ab
W
2000
1500
1000k
500
500
2000
1500
1000
500
500
2000 i
1500
1000
500
After Drying 1 Day
•Theoretical stress based
on dynamic £
^/Theoretical stress basea
on reduced E
> < [ ; I '_
\~ dynamic
reduced
J i
\ dynamic
After Drying 7 Days
L— dynamic
\
\
y* reduced
A
500 L
After Drying 28 Days
dynamic
reduced
\T
">ahnic
reduced
2 3 4 S 6 12 3
inches from Exposed Surface
February 1946
2in.Slab
— dynamic
r
r dynamic
v
^dynamic
\ (Educed
\
I
5 ' lues "• S , kt b\fb fc, and E together with appro
bstituted into Equation 20* and comp
olid curves were plotted from bl
sho* the stresses that arc obtained when the f<
imputing is used except thai the dynamic
E. The difference in th ua a
by plastic How and thus is i ..,
!asti< deformation that baa occur*
I thai for the most part the magnitude and
es after the various  I drying and for the
Besses ol specimens are about as given bj i
ire probably in error in a Fheprin
"P* d error lies in the assumption thai the i
aken Into account by using i reduced modulus of elasti
 done when Equation 20
ends oo the past stresshisi mJv M
•See J
SHRINKAGE STRESSES IN CONCRETE 385
stress at the moment. Furthermore, as explained in the section on
plastic flow, the rate of flow for an element will depend on the rate at
which the element is tending to shrink and may not be proportional to
the stress on the element. Therefore, near the drying surface, where
the stress has been relatively high from the beginning of drying, the
plastic flow will be greater and the stress will be less than that indicated.
Slightly farther inward where the stress has only recently changed from
compression to tension the resultant plastic deformation will be less
and the stress more than that indicated. The dotted curve in the one
diagram of Fig. 27 represents an attempt to show a better estimate of
the actual stress.
Reversal of stress by plastic flow. Fig. 26 indicates that eventually the
stress in the outer fiber will become negative, i.e., compressive. In all
restrainedwarping tests that were continued until equilibrium of mois
ture content was nearly reached, the moment required to prevent warp
ing decreased to zero and would have then become negative if restraint
against negative warping had been provided. This means that when a
wall dries from two opposite sides or a prism dries from all four sides,
eventually the outer shell will be in compression and the inner core will
be in tension.
Of interest in this connection is the fact that specimens of neat cement
bars have been known to break spontaneously and audibly while
resting in place in a storage rack. The explanation is that during the
early part of drying large tensile stresses developed in the outer shell.
As a result the outer shell was first permanently elongated and then
caused to fail in tension, i.e., to crack. As drying proceeded inward, the
inner core, which had not yet been stretched, tended to become shorter
than the outer shell. The cracks closed, compression developed in the
outer shell, and tension developed in the inner core. In some cases this
tension was sufficient to cause failure of the core. A specimen would
break spontaneously when failure of the inner core occurred at a section
where the outer shell was already cracked.
Investigation of properties of concrete by means of slabs or prisms drying from one side only
As the foregoing has indicated, results from prisms drying from onl\
one side have been very valuable for ascertaining in what ways the
theory of diffusion is applicable to shrinkage of concrete. They are
also valuable for investigating certain properties of concrete, especially
if used in connection with the diffusion theory. The chief advantage of
drying a prism from only one side is that it tends to warp as well as
shorten as it dries and thereby makes possible measurements not ob
tainable on prisms drying from all surfaces.
386 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE February 1946
As explained previously, the results from prisms drying from only
one side indicated a cause for concretes of higher alkali content to show a
greater tendency to crack under some conditions. Additional results in
regard to fineness of grinding and percentage of gypsum will now be
reported for the information they give to illustrate how such prisms may
be used to investigate properties of concrete.
Effect of finer grinding on plastic flow. As explained previously, in
addition to the one cement represented in Fig. 2G and 27, three other
cements were tested at the same time and in the same manner. The re
sults for the other cements were similar in most respects to those shown
in Fig. 26 and 27 for the one cement. There were some differences, how
ever. More plastic flow occurred in the specimens made with the two
finerground cements. The differences in plastic flow with fineness of
grinding in tin ' similar to and in agreement with the difl'er
bh( se cements in the restrainedshrinkage test (Fig. 25).
The effect of finer grinding on strength is similar to that produced by
r curing and therefore since longer curing decreases plastic flow we
migh that fim ag would also decrease plastic flow. How
bed in this way all Bhowed that finer grinding
stic flow for both restrainedshrinkage and restrained
m thifl laboratory indicate that this
:mcr grinding on plastic fio\i is indirect. A given quantity of
i i a1 the time ol grinding is less effective in retarding the
earl} l reactions the finer the cement. 11 Lack of proper re
tardation tctions because of insufficient gypsum results
shrinkage tendency and greater capacity for
ccording to the indications, finer grinding, if
percentage of gypsum, indirectly produces
tendency to deform inelastically.
oswered the question introduced in the discussion
by the i • plastic flow did not also reverse the
the finerground cement I ' ibly
at tl trying the coarserground cements, becaua
gth, flow more readily bul become lees pis
at the later ages when the strengths are
• talized. I nice only slight differences in the
I viate from the theory could aceount fol
d since i from the theory in many differ
possible answers to the question.
ited in Rg. 24, according to which
grind Ited in u e at the end of the curing
produced by the the
SHRINKAGE STRESSES IN CONCRETE
387
concrete to expand during curing, especially the tendency to expand
after some resistance to plastic flow had developed. Any tendency
for the interior of the freeshrinkage specimens to continue expansion
after the surface begins to dry would reduce the rate of shortening at the
beginning of the drying period. If, as seems quite probable, this tendency
is greatest for the thicker specimens with the coarserground cements,
then these specimens would shorten relatively less at the beginning of
drying than would be indicated by theory.
Effect of added gypsum. The effect of the gypsum in the cement on
the properties of the hardened concrete was observed in an investiga
tion in which 21 cements were made from the five clinker compositions
listed in the Appendix. By blending various grinds of these clinkers,
cements of different finenesses and different gypsum contents were
obtained from each clinker. Concretes (Mix C) made from these 21
cements were tested in the manner indicated previously for prisms drying
from only one side. However, in this investigation only the 3in. size of
specimen was used.
Where the C^A content of the clinker was moderate or relatively high,
an increase in SOz, content decreased shrinkage and warping and also
decreased plastic flow. Where the C 3 A content was low, an increase in
SOz had relatively little effect. According to other data obtained in this
laboratory, a still further increase in SO z would have increased the
shrinkage. 24 Representative results for a cement of high CzA content
are shown in Fig. 28, 29, and 30. As shown by Fig. 28, the maximum
warp of those specimens free to warp was reduced appreciably by in
crease in per cent of SOz The reduction in shortening with increase in
SOz agrees with that reported previously by other investigators 21  24  25 .
Fig. 29 shows that the restraint developed by the specimens restrained
against warping was in general less with the higher percentages of SOz
However, increasing the SOz from 1.5 to 2.4 per cent had only a very
small effect on the amount of restraint developed in the restrained
specimens compared to the effect on the warping of unrestrained speci
mens. The explanation is that although the increase in SOz reduced
warping it also reduced the tendency to yield under stress. The net
result is some reduction in stress but not as much as would be antici
pated from the results of the freewarping specimens.
Fig. 30 shows the effect of SOz on the factor of safety against cracking
as determined by this test of a cement of high CzA.
SUMMARY AND CONCLUSIONS
The theory that shrinkage of concrete follows the laws of diffusion
similar to those followed by the flow of heat is tested by means of specially
388 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE February 1946
jba8 uo spunod  uiej{S9^(
siuuoiiiiai ui 6uiu8j04? iun
8 8 8 8 8
o
4°
.1='
c —
5 *
u *
0)
.2 >
I ^
Q. 
>.£
21
i
ZE 2
5£ O
._
E >
• o
SHRINKAGE STRESSES IN CONCRETE
389
3.0
2.6
i ! r — i 1 —
f r ■ t r r t
7 days) j ^

Cured 7 days \
Jp. Surface 1500^^ ■
~Zs^ X / 28 days

/ s^
/ ^^C/ 28 days
^^ \ 2IOO
_
_.. II,,,
i.i,];
'
Fig. 30 Effect
of gypsum, peri
od of curing, and
fineness of grind
ing on factor of
safety against
cracking
Specimens same as
those represented in
Fig. 29.
10 1.5 2.0 2.5 30 3.5
5O3 Content of Cement, % by weight
4.0
designed experiments. According to the theory th developed in
Part 1, the shrinking and development of stress in a given coni
under given conditions of drying is consii be chai
certain constants. These constants are diffusivity of shrinkage, suri
factor, ultimate shrinkage, and Young's modulus of elasticity. Equa
tions were derived in Part 1 g I warping of prisma
versus period of drying in terms of these constants and the dimensions
of the prisms.
In Part 2 it is shown that these con m be seta
shortening of a prism as computed by the theoretical equations 1 in
good agreement with experimental values of shortening. Furthern
it is >hou 11 that by using the same constants the shortening versus period
of drying of other prisms differing in size and number of 
to drying can be predicted with fair accuracy if the difference in si
not too great. However, it is shown that the theory r modified
to take into account inelastic deformation and to permit the supposed
constants to vary with moisture id size of specimen if the
theory is to be in agreement with all results on all t> mens
of a given concrete.
The theory is used to explain various things about concrete; in I
paradoxically, it is used to explain some of th< which con<
does not perform as predicted by tin theory. The tendency of I
specimens to crack more ami shrink Less than smaller specimens and the
effect of alkali content of the cement in increasing the tendency to warp
while reducing the rate of shrinkage are explained on th the
390 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE February 1946
theory. It is shown that when a saturated specimen is dried at 50 per
cent relative humidity the stress developed would be much greater
than the strength of the concrete if it were not for the effects of plastic
flow. It is further shown that when a specimen is restrained against
deforming the restraining forces are much less than they would be if
plastic flow did not occur.
An example is given of the use of the theoretical equation in determin
ing the distribution of stresses at various times during the drying of a
specimen. In this example, consideration is given to plastic flow and to
the decrease in diffusivity of shrinkage as the specimen dries.
The restrainedshrinkage test and the restrainedwarping test are
used to determine a factor of safety against cracking for concrete under
conditions of drying and of restraint comparable to those under which
the tests are made. These tests, together with tests on freeshrinkage
and freewarping specimens, are used to measure plastic flow.
The various tests described in Part 2 and in the Appendix to Part 2
when used in conjunction with the theory given in Part 1 provide a
means for studying some of the more fundamental properties of concrete
and for predicting the performance of concrete under some conditions
in the field.
ACKNOWLEDGMENTS
This paper developed during a study of the causes and the control
of cracking of concrete. The author is. indebted to many present and
past workers in this field. E. A. Ripperger, now Lt. (jg.) U.S.N, with
the Pacific fleet, was responsible for a large part of the experimental
work reported herein. He designed (or adapted from earlier designs)
most of the special equipment used and was engaged in certain phases of
the study previous to the author's participation. All of the work was
done under the supervision of F. R. McMillan, Director of Research,
and T. C. Powers, in charge of Basic Research.
The author is particularly indebted to Mr. Powers for assistance
in preparing the manuscripts. His suggestions in regard to presentation
of material and the wording of various paragraphs have been invaluable.
The author also wishes to thank Miss Adele Scott for preparing the
diagrams and Miss Virginia Atherton for proofreading the manuscript.
SHRINKAGE STRESSES IN CONCRETE
391
APPENDIX TO PART 2
Mix proportions
Parts by Weight
Mix A
Mix B
Mix C
0.5
0.5
1.0
0.6
0.355 to 0.388
0.048
0.403 to 0.436
1.0
1.28
1.82
487
Water added for absorption
083
570
1
Elgin sand
2 43
2 97
Consistency
Consistency of Mix B with different cements was maintained fairly constant at from
5 to 6 in. of slump with a 12in. cone by varying the amount of mixing water. The
consistencies of Mixes A and C were allowed to vary with the different cements. Mix C
usually gave a slump of from 2 to 4 inches, but with some cements the slump was as
little as 1.5 inches and with others as much as 6 inches.
Materials
Cements: One cement designated M was a mixture of four brands of Type I cement,
purchased in Chicago. Its specific surface by the Wagner method was 1665 sq. cm.
per g. The other cements were prepared from five different commercial clinkers. From
each of these clinkers cements of three different finenesses, coarse, medium, and fine,
were prepared by grinding at the plant. In addition, two cements, one of low and one
of high gypsum content, were prepared from each clinker by grinding in a small lab
oratory ball mill. The purpose in preparing these five different cements from each
clinker was to make it possible to obtain any desired fineness and gypsum content by
blending different grinds of the same clinker. In referring to these cements in the text
the first number in the designation is the clinker number, the second is the Wagner
specific surface, and the third is the per cent S0 3 content by weight.
The chemical compositions of the five clinkers and of cement M are shown below.
Oxides
Cement Clinker No.
Chemical Analysis,
per cent by wt. (corr. for minor components)
20.67
23.05
27.82
5.48
4.14
1.93
2.50
4.35
1.87
65.00
64.28
65.38
1.31
1.36
1.75
0.19
0.03
0.17
0.85
1.05
0.26
2.71
0.73
0.23
0.30
0.05
0.05
0.40
0.17
0.22
Cement
M
SiOi
AhOa...
Fe20 2
Combined CaO
MgO
S0 3
Loss on Ign. . . .
Free CaO
Na 2
K 2
Compounds
C3S
C2*
I \
< YU ,
21.54
6.52
1.56
64 . 32
2,17
0.41
0.15
0.98
0.17
0.16
50.73
23.49
14.72
\ 75
22.56
5.00
2.48
64 06
3.35
0.20
0.37
nil
1.13
0.44
Computed Compound Composition
per cent by wt.
66.57
52.37
38.58
51.61
9.05
26.58
50.66
25.75
10.29
3 61
1.95
9.06
7.61
13.24
5.69
7.55
21.25
5.98
2.69
62.56
3.04
1.75
1.13
0.79
0.28
0.63
44.15
27.62
11.30
^ I'.i
392 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE February 1946
Aggregate: Sand and gravel were from Elgin, Illinois. The gravel was screened
to pass a %in, sieve and be retained on a Xo. 4 sieve.
The sand was graded as folL
Sieve Per Cent
Xo. Bine
100 a
15
28 4:<
14
82
4 100
Pulverized Silica: The silica was from the same source as standard Ottawa sand, but
ground to cement fineness. Its specific surface b) the airpermeability method was 3200
sq. cm. per g.
Procedure
With some exceptions the procedure in preparing and testing the specimens was as
follows:
Preparing the specimens: The materials were mixed in a powerdriven opentub
mixer. The fractions of the varioussized aggregates and grinds of cement were weighed
and placed in the tub. The mixing schedule was: mix ] _> minute dry, 2 minutes wet,
mix 2 minutes. (Special tests showed that the grinds of cement were
adequately blended in }/> minute of dry mixing in the proence of sand.)
The freshly mixed concrete was placed in steel molds and consolidated by light
vibration by placing molds on a platformtype vibrator. Covers were fastened on the
molds but not made watertight. Each mold was equipped with restraining bars,
gage inserts, etc., the details of assembly depending upon the tests to be made.
The molds and contents were stored under water at 74 F for one day. The molds
were then stripped and the specimens returned to water at 76 =«= 1 F in a covered tank
where they were left for one hour. The specimens were then removed one at a time,
dried with a cloth, and all initial measurements of length, deflection, and weight were
made. All dimension measurements were made within 30 seconds after removal from
water, and weighings were made as soon thereafter as practical. The specimens were
then returned to the 76 F curing tank.
One day before the end of the curing period the specimens were removed for sealing
of certain surfaces against the loss of moisture. The surfaces to be sealed were wiped
with a cloth and then allowed to airdry until the surface just changed color. During
.me. scheduled measuren. usually made. After the color change and
before any appreciable loss of moisture by evaporation, one coat of black, quickdrying
brushing lacquer was applied. After the lacquer had dried a few minutes, the 
■d with a cloth and one coat of hot paraffin was applied to the lacquered
surface. While a second coat of paraffin was being applied, one thickness of t *
paper of appropriate shape wa pressed into the still soft parafiin somewhat in the
manner in which a paper hanger applies wallpaper. Next, a final heavy coat of paraffin
ipplied. The layer of paper helped to eliminate pin holes. (When only the ends
of prisms were Bealed, the paper and final coat were omitted.)
r the specified surfa Jed. the unsealed surfaces and exposed steel parts
were cleaned, vaseline was applied to the steel parts, and the specimen was returned to
the curing tank for an additional day of curing. At the end of the curing period the
specimens were transferred from the curing tank to a room maintained at 76 ± 1 F and
SHRINKAGE STRESSES IN CONCRETE
393
50 ± 2 per cent relative humidity. At this time the vaseline was removed from the
steel parts and measurements for the beginning of the drying period taken.
Testing the specimens: The testing was considered to have begun in most of the
tests with the beginning of the drying period. The specimens may be divided into
classes, according to the tests made, as follows:
Free shrinkage specimens were measured for length changes, weight losses and resonant
frequency of vibration. Reference plugs were cast in the ends of the specimens for the
lengthchange readings. They were hexhead cap screws arranged to give the desired
gauge length. Ordinarily these were single plugs centrally located in the ends. But
for those prisms that represented slabs drying from two opposite sides the arrange
ment was as shown in Fig. 31.
o o o
>i" \ii" \ti"\ f i" ]
6" 
o
o
o
sea/ed^\
6"
h*H
1
Z"
o
2i»
o
.__
J
o
H'
o
2*
FH
Fig. 31 — End views showing arrangement of gage plugs in specimens representing 2in,, 6
in. and 12in. slabs
The gage plugs for these specimens were Y% in. bolts 4 in. long.
Restrained shrinkage specimens were partially restrained against shrinkage by steel
restraining bars. The arrangement of the bars in those specimens that represented
slabs drying from two opposite sides was the same as that shown in Fig. 31 for gage plugs
except that only two bars 3 inches apart were cast in the 2x6in. specimens. Square
specimens that are permitted to dry from all four sides and that are partially restrained
by one centrally located bar are used in routine testing of resistance to cracking. Further
details in regard to restraining bars and the measurements for change in length of the
restraining bar are shown in Fig. 32 and 33 as well as in Fig. 23 body of the 1 1
If the restrainedshrinkage specimen did not crack spontaneously before the maximum
restraining force had been developed, additional increments of load were applied as
shown in Fig. 33 until failure was produced.
Free warping specimens were measured for deflection over a 32in. span. Most of
these specimens were also measured for length change, weight loss and resonant fre
394 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE February 1946
quency. Some of the free warping specimens made in the early part of the work were
not equipped for lengthchange measurements. Cross sections of free warping speci
mens are shown in Fig. 34. See Fig. 35.
Restrained warping specimens companion to the free warping specimens were loaded,
as shown by Fig. 36, so as to prevent warping between the loaded quarterpoints. Read
ings of the load required were taken periodically and, after the maximum restraint had
developed, the load necessary to produce failure was determined.
Fig. 32a (left) Comparator
Fig. 32b (above) ^Restraining Bar
Measurements of shrinkagestress are made whether cracking occurs or not. The
comparator (in Fig. 32a) is used to measure the changes in length of the steel restrain
ing bar that result from the strains placed on it by the concrete. From these measure
ments Hid the known properties of steel, the average stress in the concrete is computed
Important details of the restrainedshrinkage type of specimens are shown in Fig. 32b
(See also Fig. 23 in the tea thai the o an grip the bar only in the end
region; contact intl 2 tion is prevented by a thick layer of rubber.
SHRINKAGE STRESSES IN CONCRETE
395
Fig. 33a
Fig. 33b
This method of test has been used to measure shrinkagestress in specimens as wide
as 12 in. and containing as many as four restraining bars.
Those specimens that do not fail under shrinkagestress alone are given additional
stress with the machines shown in Fig. 33a and 33b. The machine at the left (Fig. 33a)
is used for most of the specimens; that at the right Fig. 33b is used if the capacity of the
other one is exceeded and if the specimen contains more than one bar. The load is
applied to the bar, and the extension of the bar at the time the concrete fails is determined
by the straingage shown in the pictures. The net amount of load on the concrete at
failure is computed from the magnitude of the load and the straingage reading. The
factor of safety is the ratio of the load on the concrete at failure to the maximum load
represent od by the restraint against shrinkagi .
"Sonic" testing: Most of the "unrestrained" specimens are tested periodically for
frequency of vibration with the apparatus shown in Fig. 35a. Young's modulus is
calculated from the resonant frequencies.
ill
396 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE February 1946
sealed — \
r
r
j"
\*\
Fig. 34 — Cross section of
free warping specimens.
Fig. 35a Sonic testing
Fig. 35b— Warping
measurement
SHRINKAGE STRESSES IN CONCRETE
397
Fig. 36a (above)
Fig. 36b (left)
Restrained warping
398 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE February 1946
Warping, lengthchange, and moisturecontent: On specimens like that shown in Fig.
35a, the change in length is measured with the comparator shown in Fig. 32 and the
warping is measured with the curvaturegage shown resting on a specimen in Fig. 35b.
Results are correlated with concomitant changes in moisturecontent.
Warping due to swelling: Previously dried specimens are placed, uncoatod side down,
in a trough of water as shown Fig. 35b or ( bey are exposed to saturated air. The result
ing warping due to absorption at one surface is more rapid than that due to shrinkage
and can ^ive rise to larger stresses.
The specimens shown Fig. 36 are supported only at the ends. They are coated on all
hut the bottom side and therefore as t bey dry they tend to bow upward. This tendency
i> opposed b) th< shackles ai tin quarterpoints which are connected to a lever system
below, one for each specimen. The levers are held by the finethread screwadjustment
seen best in Fig. 36b. The screws are turned dow award until the force is just sufficient
to prevent warping as indicated by the curvaturegage shown in both pictures. This
i be moved from specimen to specimen.
The force on the lever is measured periodically by finding the weight (bucket of shot)
that will just hold the specimen in the position maintained by the screw. The force
required reaches a maximum and ilnn n cedes slowly to zero as drying continues. When
the maximum is reached, i he sp< cimens are loaded to failure. The ratio of the maximum
>:nn\ to pn ping to that required for failure Lb the factor of safety.
the factor of safety against cracking of Blabs of twice
of the spe<  i quail} from both sides.